Probability is a chapter in the CBSE Class 10 Mathematics syllabus from Mathematics. This chapter hub brings together revision notes, practice questions, worksheets, flashcards, formula sheet to help students learn, practice, and revise Probability effectively.

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Probability

NCERT Class 10 Mathematics Chapter 14: Probability (Pages 202–217)

Class 10 CBSE hub Mathematics chapters Mathematics

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Summary of Probability

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Probability at a Glance

Board

CBSE

Class

Class 10

Subject

Mathematics

Book

Mathematics

Chapter

Pages

202–217

Resources

7 study resources

Probability Summary

Probability is the measure of how likely an event is to occur. Understanding probability helps us make informed decisions based on the likelihood of various outcomes. In this chapter, we learn about theoretical probability, which is based on the assumption that all outcomes are equally likely. For instance, when tossing a fair coin, there are two possible outcomes: heads or tails, each with an equal chance of one half. Similarly, when throwing a fair die, the outcomes are the numbers one through six, all equally likely. We also differentiate between equally likely outcomes and those that are not. For example, drawing a ball from a bag containing four red balls and one blue ball shows that the probability of drawing a red ball is greater than that of drawing a blue ball. The definition of probability considers the number of favorable outcomes to the total outcomes, allowing for straightforward calculations. Additionally, we explore concepts such as complementary events—where the sum of the probabilities of two outcomes equals one—and the concepts of certain events and impossible events, where probabilities are one and zero, respectively. Throughout the chapter, we analyze various examples and exercises that reinforce the application of probability in real-life situations, such as drawing from a deck of cards or selecting students from a class. We conclude by extending the idea of probability to continuous outcomes, where we examine cases where outcomes fall within ranges, helping us to understand how probability applies in broader contexts. This knowledge is valuable across fields like statistics, gaming, science, and everyday decision-making.

Probability Revision Guide

Download the Probability revision guide with key points, summaries, and quick revision notes for CBSE Class 10 Mathematics.

Key Points

Definition of Probability

The probability of an event E is defined as P(E) = Number of favorable outcomes / Total outcomes.

Fair Coin Toss

A fair coin has two outcomes (Head or Tail) with equal probability of 0.5 each.

Equally Likely Outcomes

Events with the same chance of occurring are termed equally likely, affecting probability calculations.

Theoretical Probability

Theoretical probability is based on reasoning or assumptions rather than repeated experiments.

Empirical Probability

Calculated as P(E) = Number of times E occurs / Total trials, useful for experiments.

Complementary Events

If P(E) is the probability of event E, then the probability of 'not E' is P(not E) = 1 - P(E).

Sum of Probabilities

The sum of probabilities of all possible outcomes in an experiment equals 1.

Elementary Events

Events that consist of one single outcome are called elementary events (e.g., tossing a Tail).

Probability of Impossible Events

An event that cannot happen has a probability of 0, e.g., getting a 7 from a single die toss.

Probability of Certain Events

An event that is sure to happen has a probability of 1, e.g., rolling a number ≤ 6 on a die.

Calculating Dice Probabilities

For a single die, P(E) calculations involve counting the desired outcomes among 6 total outcomes.

Drawing Balls from a Bag

If a bag has differing colored balls, calculate probability by dividing favorable colors by total balls.

Playing Cards Probability

For a 52-card deck, P(drawing an Ace) = 4/52 = 1/13, P(not drawing an Ace) = 48/52 = 12/13.

Tossing Two Coins

When tossing two coins, calculate outcomes (HH, HT, TH, TT) and find P(at least one Head) = 3/4.

Events with Multiple Outcomes

Complex events can combine several outcomes, necessitating additional calculations for probabilities.

Real-World Applications

Probability is used across fields like economics, genetics, and weather forecasting to predict outcomes.

Law of Large Numbers

As the number of trials increases, empirical probabilities converge on theoretical probabilities.

Probabilities in Sports

Probabilities are often applied in sports predictions, influencing betting and strategy decisions.

Expected Value Concept

The expected value summarizes the average outcome of a probabilistic event over time or trials.

Understanding Graphical Representations

Diagrams can effectively illustrate probabilities, such as pie charts or probability trees, enhancing comprehension.

Overcoming Misconceptions

Clarify that past results do not influence future independent trials (e.g., coin tosses).

Probability Practice Questions & Answers

Practice important questions and exam-style problems from Probability. These questions cover key topics from the CBSE Class 10 Mathematics syllabus.

How to practice: Start with the questions below to test your understanding of Probability. Use the revision guide to review concepts you find difficult, then come back and retry the questions for better retention.

Which of the following is an example of an elementary event?

Single Answer MCQ

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Which statement about impossible events is true?

Single Answer MCQ

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In a single trial of tossing a fair coin, what type of events are 'heads' and 'tails'?

Single Answer MCQ

Q-00174475

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If P(A) + P(A') = 1 for any event A, what does A' represent?

Single Answer MCQ

Q-00174476

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Which event can be classified as a sure event?

Single Answer MCQ

Q-00174477

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If you flip a fair coin twice, what are the possible outcomes that contain at least one tail?

Single Answer MCQ

Q-00174478

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What is the probability of getting heads when a fair coin is tossed?

Single Answer MCQ

Q-00174479

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Which of the following probability values is incorrect?

Single Answer MCQ

Q-00174480

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View all 206 Probability questions

When a coin is tossed, which of the following outcomes is considered favorable if we want to get tails?

If a fair coin is tossed twice, what is the probability of getting at least one head?

What is the probability of tossing a fair coin three times and getting exactly two heads?

In a single toss of a fair coin, what is the probability of not getting heads?

Which of the following statements about a fair coin is true?

What is the total number of possible outcomes when a fair coin is tossed twice?

If a coin is tossed until heads appears, what is the probability that it will take exactly four tosses to get heads?

If you toss a coin 100 times, how many heads do you expect to obtain?

What is the probability of getting tails if you have already gotten heads on the first toss?

After tossing a fair coin 10 times, if you obtain heads 6 times, how would you calculate the probability of getting heads on the next toss?

Which of the following represents a type of probability that refers to outcomes in a theoretical sense?

In a fair coin toss, if you want to find the probability of exactly one head in three tosses, what is the method to find it?

What is the probability of rolling a fair die and getting a number greater than 4?

What is the probability of tossing a fair coin and getting heads twice in a row?

What is the total number of equally likely outcomes when tossing a fair coin?

When rolling a fair six-sided die, what is the probability of rolling an even number?

In a bag of 10 balls (3 red, 2 blue, 5 green), what is the probability of picking a red ball?

How many equally likely outcomes are there when spinning a fair spinner divided into 4 equal sections?

If you randomly select a card from a standard deck of 52 cards, what is the probability of drawing a heart?

What is the probability of rolling a sum of 7 with two six-sided dice?

What is the empirical probability of an event?

In a lottery with 3 winning numbers chosen from 10 total numbers, how many equally likely outcomes are there for the winning combination?

Which scenario is best suited for calculating empirical probability?

If the probability of an event is 1/5, what is the probability that the event does not occur?

What defines the theoretical probability of an event?

When flipping two coins, what is the probability of getting at least one head?

Which of the following statements about empirical and theoretical probability is true?

In a group of 12 students, how many ways can you select 2 students?

In a fair coin toss, what is the theoretical probability of getting heads?

If a spinner is divided into 5 equal sections, with colors red, blue, green, yellow, and orange, what is the probability of landing on blue?

If an experiment results in 20 successes out of 100 trials, what is the empirical probability of success?

If there are 20 students in a class and each student has an equal chance of being selected as a class representative, what is the probability of selecting a specific student?

Theoretical probability assumes which of the following?

What is the probability of drawing a queen from a standard deck of playing cards?

An event occurs 30% of the time during trials. If we conduct 50 trials, how many times do we expect the event to occur theoretically?

In which situation would you prefer theoretical probability over empirical probability?

If a dice is rolled once, what is the theoretical probability of rolling a number greater than 4?

What is a common pitfall when interpreting empirical probability?

If an event has a theoretical probability of 0.4, what is the empirical probability if 40 successful outcomes are observed in 200 trials?

Pierre Simon Laplace is known for which of the following contributions?

What is the consequence of using a biased die in empirical probability measurement?

What is the classical probability of an event defined as?

If a coin is tossed, what is the classical probability of getting tails?

How is classical probability of getting a specific face from a die calculated?

If an event has 3 favorable outcomes from 10 possible outcomes, what is its classical probability?

What was one of the first significant contributions to probability theory?

Which of the following statements reflects a common misconception in probability?

What would be the probability of rolling an even number on a fair six-sided die?

In a game where a player picks a card from a standard deck, what is the probability of picking an Ace?

If three coins are tossed, what is the probability of getting at least one tail?

What is the meaning of 'favorable outcomes' in classical probability?

If an event has a classical probability of 0, what does that signify?

Which statement correctly defines an event having a classical probability close to 1?

What is the probability of rolling a sum of 7 with two dice?

In a bag of 10 balls, 4 are red and 6 are blue. What is the probability of randomly selecting a blue ball?

What's the probability of drawing a heart from a standard deck of 52 playing cards?

If the probability of raining tomorrow is 0.3, what is the probability that it will not rain?

A coin is tossed three times. What is the probability of getting exactly two heads?

A box contains 5 red, 3 blue, and 2 green marbles. What is the probability of picking a red marble?

In a game, you roll a die. What is the probability of rolling a prime number?

If the probability of event A occurring is 0.4 and the probability of event B occurring is 0.5, what is the probability of both A and B occurring if they are independent?

What is the probability of getting at least one tail when flipping two coins?

In a lottery, the probability of winning is 1 in 1000. What is the probability of losing?

What is the probability of drawing one king from a standard deck of cards?

The probability that a student passes an exam is 0.75. What is the probability that the student fails?

If the probability of drawing a red card is 0.5, and you draw two cards without replacement, what is the probability that both are red?

What is the probability of selecting a number less than or equal to 3 when choosing from the set {1, 2, 3, 4, 5, 6}?

What is the probability of rolling a 6 on a fair die followed by a 5 on another fair die?

Two dice are thrown simultaneously. The probability of getting a sum of 7 is:

Raghav has a collection of balls of different colours. He has a total of 35 balls in his basket out of which seven are black in colour and eight are yellow in colour. Out of remaining balls, some are white and the rest are red. Find the probability of drawing a ball at random from the basket which is either a black or a white ball.

Probability of getting an irrational number at random from the numbers √3, √4, ∛9, ∛8, √5, 0, 4 2/3 is:

A card is drawn from a well shuffled deck of 52 cards. The probability that it is not a diamond card is:

Assertion (A): The probability of an event cannot be 1/0.9. Reason (R): 0 ≤ P(E) ≤ 1 for an event E.

Two dice are rolled together. Find the probability that in the obtained outcomes one number is twice the another.

Two dice are rolled together. Find the probability that both the numbers obtained are greater than 4.

For an event E, P(E) + P(Ē) = x, then the value of x^2 - 3 is:

Assertion (A): The probability that a leap year has 53 Mondays is 2/7. Reason (R): The probability that a non-leap year has 53 Mondays is 5/7.

Two dice of different colours are thrown at the same time. Write down all the possible outcomes. What is the probability that: (i) same number appears on both the dice; (ii) different numbers appear on both the dice?

In a random experiment of throwing a die, which of the following is a sure event?

A lot consists of 200 pens of which 180 are good and the rest are defective. A customer will buy a pen if it is not defective. The shopkeeper draws a pen at random and gives it to the customer. What is the probability that the customer will not buy it? Another lot of 100 pens containing 80 good pens is mixed with the previous lot of 200 pens. The shopkeeper now draws one pen at random from the entire lot and gives it to the customer. What is the probability that the customer will buy the pen?

Meena calculates that the probability of her winning the first prize in a lottery is 0.08. If total 800 tickets were sold, the number of tickets bought by her, is

Which of the following can not be the probability of an event?

Two dice are rolled together. The probability of getting an outcome (x, y) where x > y, is

A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is 3/5, then find the number of yellow balls.

The probability for a randomly selected number out of 1, 2, 3, 4, ..., 25 to be a composite number is:

Two dice are thrown at the same time. Determine the probability that (i) sum of the numbers on the two dice is 5, and (ii) difference of the numbers on the two dice is 3.

The probability for a randomly selected number out of 1, 2, 3, 4, ..., 25 to be a composite number is:

Two dice are thrown at the same time. Determine the probability that (i) sum of the numbers on the two dice is 5, and (ii) difference of the numbers on the two dice is 3.

The probability for a randomly selected number out of 1, 2, 3, 4, ..., 25 to be a composite number is:

Two different dice are thrown together. Find the probability that the numbers obtained have even sum.

Two different dice are thrown together. Find the probability that the numbers obtained have even product.

Two dice are rolled together. The probability that the sum of the numbers obtained is divisible by 6 is:

A group of friends wanted to play cards with two identical packs together. While shuffling the cards, three cards are dropped. Rest of the cards are shuffled and one card is drawn at random. Assuming that the dropped cards were a queen of hearts, a ten of spades and an ace of clubs, find the probability that the drawn card is a face card.

In the same card situation with two identical packs and dropped cards queen of hearts, ten of spades and ace of clubs, find the probability that the drawn card is either a king or a queen.

Do you think that the probability of getting a queen was higher if none of the cards were dropped? Justify your answer.

Find the probability that the drawn card is a jack. Compare it with the probability when none of the cards were dropped. In which case is the probability of getting a jack higher?

Three coins are tossed together. The probability of getting exactly one head is:

Two dice are rolled together. The probability that the sum of the numbers obtained is divisible by 6, is:

Find the probability that the drawn card is either a king or a queen.

Do you think that the probability of getting a queen was higher if none of the cards were dropped? Justify your answer.

Find the probability that the drawn card is a jack. Compare it with the probability when none of the cards were dropped. In which case is the probability of getting a jack higher?

A die is thrown once. Probability of getting a number other than 3 is:

Assertion (A): The probability that a leap year has 53 Mondays is 2/7. Reason (R): The probability that a non-leap year has 53 Mondays is 5/7. Choose the correct option.

Two dice of different colours are thrown at the same time. Write down all the possible outcomes. What is the probability that: (i) same number appears on both the dice? (ii) different number appears on both the dice?

A die is thrown once. Probability of getting a number other than 3 is:

Assertion (A): The probability that a leap year has 53 Mondays is 2/7. Reason (R): The probability that a non-leap year has 53 Mondays is 5/7.

Two different coins are tossed simultaneously. What is the probability of getting at least one head?

Two different coins are tossed simultaneously. What is the probability of getting at most one tail?

Two different coins are tossed simultaneously. What is the probability of getting a head and a tail?

A bag contains 3 red, 4 white and 7 green balls. A ball is drawn at random. The probability that the ball drawn is not of red colour is:

In an experiment of throwing a pair of dice, the probability of not getting a doublet is:

The total number of outcomes in the experiment of simultaneous throw of three dice is:

A game of chance consists of spinning a wheel which comes to rest at one of the numbers from 1 to 10 with equal probabilities. What is the probability that the wheel stops at an odd number less than 9?

A game of chance consists of spinning a wheel which comes to rest at one of the numbers from 1 to 10 with equal probabilities. What is the probability that the wheel stops at a prime number greater than 2?

A game of chance consists of spinning a wheel which comes to rest at one of the numbers from 1 to 10 with equal probabilities. What is the probability that the wheel stops at a multiple of 4?

The total number of outcomes in the experiment of simultaneous throw of three dice is:

A game of chance consists of spinning a wheel which comes to rest at one of the numbers from 1 to 10 with equal probabilities. What is the probability that the wheel stops at a multiple of 4?

The total number of outcomes in the experiment of simultaneous throw of three dice is:

A game of chance consists of spinning a wheel which comes to rest at one of the numbers from 1 to 10 with equal probabilities. What is the probability that the wheel stops at: (i) a prime number greater than 2; (ii) an odd number less than 9; (iii) a multiple of 4?

A pair of dice is thrown simultaneously. Let E denote the event that “The sum of numbers obtained on both dice is at least 9.” The number of outcomes in favour of event E is:

A box contains 6 blue, 4 white and 8 red marbles. A marble is drawn at random from this box. Find the probability that the marble so drawn is white or red.

A box contains 6 blue, 4 white and 8 red marbles. A marble is drawn at random from this box. Find the probability that the marble so drawn is white.

A box contains 6 blue, 4 white and 8 red marbles. A marble is drawn at random from this box. Find the probability that the marble so drawn is not red.

A pair of dice is thrown simultaneously. Let E denote the event that “The sum of numbers obtained on both dice is at least 9.” The number of outcomes in favour of event E is:

A box contains 6 blue, 4 white and 8 red marbles. A marble is drawn at random from this box. Find the probability that the marble so drawn is white or red.

A box contains 6 blue, 4 white and 8 red marbles. A marble is drawn at random from this box. Find the probability that the marble so drawn is white.

A box contains 6 blue, 4 white and 8 red marbles. A marble is drawn at random from this box. Find the probability that the marble so drawn is not red.

A pair of dice is thrown simultaneously. Let E denote the event that “The sum of numbers obtained on both dice is at least 9.” The number of outcomes in favour of event E is:

A box contains 6 blue, 4 white and 8 red marbles. A marble is drawn at random from this box. Find the probability that the marble so drawn is white.

A box contains 6 blue, 4 white and 8 red marbles. A marble is drawn at random from this box. Find the probability that the marble so drawn is white or red.

A box contains 6 blue, 4 white and 8 red marbles. A marble is drawn at random from this box. Find the probability that the marble so drawn is not red.

Probability of getting an irrational number at random from the numbers √3, √4, 3√9, 3√8, √5, 0, 4^(2/3) is:

A card is drawn from a well shuffled deck of 52 cards. The probability of getting an ace or a ten is:

Assertion (A): The probability of an event cannot be 1/0.9. Reason (R): 0 ≤ P(E) ≤ 1 for an event E.

Two dice are rolled together. Find the probability that the product of the numbers obtained is 6.

Two dice are rolled together. Find the probability that the sum of the numbers obtained is 10.

A card is drawn from a well shuffled deck of 52 cards. The probability of getting an ace or a ten is:

Probability of getting an irrational number at random from the numbers 3, 4, cube root of 9, cube root of 8, sqrt(5), 0, 4^(2/3) is:

Assertion (A): The probability of an event can not be 1/0.9. Reason (R): 0 <= P(E) <= 1 for an event E.

Two dice are rolled together. Find the probability that at least one of the numbers obtained is a multiple of 3.

The probability of getting sum greater than 10, when two dice are rolled together, is

A bag contains some red and some white balls. A ball is drawn at random from the bag. If the probability of getting a red ball is 2/7, then the probability of getting a white ball is

A card is drawn from a well-shuffled deck of 52 playing cards. The probability of getting a queen of spade is

Find the probability that a number selected at random from the numbers 30, 31, 32, 33, ..., 60 is a prime number.

Find the probability that a number selected at random from the numbers 30, 31, 32, 33, ..., 60 is a multiple of 6.

Slips of letters of the word 'BACKGROUND' are put in a bowl and thoroughly mixed. One slip is picked up at random. Find the probability that the picked up slip's letter is a vowel.

Slips of letters of the word 'BACKGROUND' are put in a bowl and thoroughly mixed. One slip is picked up at random. Find the probability that the picked up slip's letter is present in the word 'BALL'.

The probability of getting sum greater than 10, when two dice are rolled together, is

A bag contains some red and some white balls. A ball is drawn at random from the bag. If the probability of getting a red ball is 2/7, then the probability of getting a white ball is

A card is drawn from a well-shuffled deck of 52 playing cards. The probability of getting a queen of spade is

Find the probability that a number selected at random from the numbers 30, 31, 32, 33, ..., 60 is a prime number.

Find the probability that a number selected at random from the numbers 30, 31, 32, 33, ..., 60 is a multiple of 6.

Slips of letters of the word 'BACKGROUND' are put in a bowl and thoroughly mixed. One slip is picked up at random. Find the probability that the picked up slip's letter is a vowel.

Slips of letters of the word 'BACKGROUND' are put in a bowl and thoroughly mixed. One slip is picked up at random. Find the probability that the picked up slip's letter is present in the word 'BALL'.

A bag contains some red and some white balls. A ball is drawn at random from the bag. If the probability of getting a red ball is 2/7, then the probability of getting a white ball is

Three coins are tossed together. The probability of getting exactly two tails is

A card is drawn from a well-shuffled deck of 52 playing cards. The probability of getting a queen of spade is

Find the probability that a number selected at random from the numbers 30, 31, 32, 33, ..., 60 is a prime number.

Find the probability that a number selected at random from the numbers 30, 31, 32, 33, ..., 60 is a multiple of 6.

Slips of letters of the word 'BACKGROUND' are put in a bowl and thoroughly mixed. One slip is picked up at random. Find the probability that picked up slip's letter is a vowel.

Slips of letters of the word 'BACKGROUND' are put in a bowl and thoroughly mixed. One slip is picked up at random. Find the probability that picked up slip's letter is present in the word 'BALL'.

A card is drawn from a packet of 50 identical cards numbered from 1 to 50. The probability of drawing a number which is a perfect square, is:

The probability of getting a sum of 7, when two dice are thrown simultaneously, is:

Assertion (A): The probability that the date of birth of a man is in the month of June is 1/12. Reason (R): There are 12 months in a year.

All the red face cards are removed from a pack of 52 playing cards. The remaining cards are well shuffled and then a card is drawn at random. Find the probability of getting a red card.

All the red face cards are removed from a pack of 52 playing cards. The remaining cards are well shuffled and then a card is drawn at random. Find the probability of getting a king or queen.

Meena calculates that the probability of her winning the first prize in a lottery is 0.08. If total 800 tickets were sold, the number of tickets bought by her, is

Which of the following can not be the probability of an event?

Two dice are rolled together. The probability of getting an outcome (x, y) where x > y, is

A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is 3/5, then find the number of yellow balls.

A card is drawn at random from a well shuffled deck of 52 playing cards. The probability that it is either a ten or a king is

Meena calculates that the probability of her winning the first prize in a lottery is 0.08. If total 800 tickets were sold, the number of tickets bought by her is

Which of the following can not be the probability of an event?

A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is 3/5, then find the number of yellow balls.

Which of the following can not be the probability of an event?

A card is drawn at random from a well shuffled deck of 52 playing cards. The probability that it is either a ten or a king is

Meena calculates that the probability of her winning the first prize in a lottery is 0.08. If total 800 tickets were sold, the number of tickets bought by her, is

A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is 3/5, then find the number of yellow balls.

Two dice are rolled together. The probability that sum of the numbers obtained is at most 12, is

Assertion (A): If probability of happening of an event is 0.2p, p > 0, then p can’t be more than 5. Reason (R): P(Ē) = 1 – P(E) for an event E.

A bag contains 30 balls out of which m number of balls are blue in colour. Find the probability that a ball drawn at random from the bag is not blue.

A bag contains 30 balls out of which m number of balls are blue in colour. If 6 more blue balls are added in the bag, then the probability of drawing a blue ball will be 5/4 times the probability of drawing a blue ball in the first case. Find the value of m.

Two different dice are rolled together. The probability that both the obtained numbers are less than 4 is

Assertion (A): If probability of happening of an event is 0.2p, p > 0, then p can't be more than 5. Reason (R): P(Ē) = 1 - P(E) for an event E.

If 6 more blue balls are added in the bag, then the probability of drawing a blue ball will be 5/4 times the probability of drawing a blue ball in the first case. Find the value of m.

A bag contains 30 balls out of which m balls are blue. Find the probability that a ball drawn at random from the bag is not blue.

Two different dice are rolled together. The probability that both the obtained numbers are less than 4, is

Assertion (A): If probability of happening of an event is 0.2p, p > 0, then p can’t be more than 5. Reason (R): P(Ē) = 1 − P(E) for an event E.

A bag contains 30 balls out of which m number of balls are blue in colour. Find the probability that a ball drawn at random from the bag is not blue.

If 6 more blue balls are added in the bag, then the probability of drawing a blue ball will be 5/4 times the probability of drawing a blue ball in the first case. Find the value of m.

A card is selected at random from a deck of 52 playing cards. The probability of it being a red face card is:

Assertion (A): The probability of selecting a number at random from the numbers 1 to 20 is 1. Reason (R): For any event E, if P(E) = 1, then E is called a sure event.

Two dice are thrown at the same time. Determine the probability that the difference of the numbers on the two dice is 2.

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Probability Practice Worksheets

Download and practice Probability worksheets to improve problem-solving accuracy and speed for CBSE Class 10 Mathematics exams.

Probability - Practice Worksheet

This worksheet covers essential long-answer questions to help you build confidence in Probability from Mathematic for Class 10 (Mathematics).

Practice

Questions

Define theoretical probability and explain its significance with real-life examples.

Theoretical probability is defined as the ratio of the number of favorable outcomes to the total number of equally likely outcomes. It is given by the formula P(E) = Number of outcomes favorable to E / Total number of possible outcomes. The significance of theoretical probability lies in its ability to estimate outcomes without the need for actual experiments. For instance, when tossing a fair coin, the theoretical probability of getting heads is 1/2 since there are two equally likely outcomes: heads and tails. Similarly, when rolling a die, the probability of getting a 3 is 1/6, as there are six possible outcomes. This allows for predictions in games, finance, and various other applications.

Explain the concept of complementary events with examples and how it helps in calculating probabilities.

Complementary events are pairs of events where one event includes all the outcomes of the experiment that are not part of the other event. If E is an event, then its complement, denoted as E', includes all outcomes not in E. The probability of complementary events always sums to 1, represented as P(E) + P(E') = 1. For example, if the probability of getting heads when flipping a coin is P(H) = 1/2, then the probability of getting tails is P(T) = 1/2. This means P(H) + P(T) = 1, confirming that these are complementary events. This concept simplifies calculations and ensures accurate predictions in probabilistic scenarios.

Calculate the probability of drawing a king from a standard deck of 52 playing cards.

A standard deck of playing cards consists of 52 cards, which includes 4 kings (one from each suit: hearts, diamonds, clubs, spades). To find the probability of drawing a king, we use the formula for probability: P(King) = Number of favorable outcomes / Total number of possible outcomes. Here, the number of favorable outcomes is 4 (the kings), and the total outcomes is 52. Thus, P(King) = 4/52 = 1/13. Therefore, the probability of drawing a king from a deck is approximately 0.0769 or 7.69%. This basic probability calculation is useful in games involving cards.

What is the probability of rolling a number greater than 4 on a fair six-sided die?

The possible outcomes when rolling a fair six-sided die are 1, 2, 3, 4, 5, and 6. The event of rolling a number greater than 4 includes the favorable outcomes: 5 and 6. The probability can be calculated using the formula P(E) = Number of favorable outcomes / Total number of possible outcomes. Here, the favorable outcomes (greater than 4) are 2 (5 and 6), and the total possible outcomes are 6. Therefore, P(Greater than 4) = 2/6 = 1/3. This outcome helps in understanding chances in games like craps or any rolling dice scenario.

In a bag containing 5 red balls and 3 green balls, what is the probability of drawing a green ball?

To find the probability of drawing a green ball from the bag, we first identify the total number of balls. There are 5 red and 3 green balls, leading to a total of 8 balls. The number of favorable outcomes for drawing a green ball is 3. Hence, using the probability formula P(E) = Number of favorable outcomes / Total number of possible outcomes, we find P(Green) = 3/8. This calculation can be particularly useful in probability theory, showcasing how to evaluate chances based on specific conditions in real-life situations.

How does the law of large numbers apply to empirical probability? Provide examples.

The law of large numbers states that as the number of trials increases, the empirical probability will converge to the theoretical probability. This principle asserts that if an event is repeated a large number of times, the average of the outcomes will be close to the expected probability. For example, if we toss a coin 1000 times, we expect approximately 500 heads and 500 tails, closely aligning with the theoretical probability of 1/2 for each outcome. Similarly, if we roll a die many times, the frequency of each number will approach 1/6. This concept is foundational for practical applications in statistics, gaming, and risk assessments.

Discuss the meaning and significance of an impossible event in probability.

An impossible event is an event that cannot occur, represented by a probability of 0. For instance, when rolling a standard six-sided die, the probability of rolling a number 7 is impossible because the die only has numbers from 1 to 6. Hence, P(rolling a 7) = 0. The significance of recognizing impossible events helps in setting realistic expectations and understanding the fundamental limits within probability theory. Knowing the boundaries of possible outcomes is crucial when analyzing experiments and making forecasts in various fields like finance, marketing, and scientific research.

What is the probability of selecting a red marble from a jar containing 2 red, 4 blue, and 3 green marbles?

To calculate the probability of selecting a red marble, first determine the total number of marbles in the jar. The total is 2 red + 4 blue + 3 green = 9 marbles. The number of favorable outcomes, which is selecting a red marble, is 2. Using the probability formula P(E) = Number of favorable outcomes / Total number of possible outcomes, we have P(Red) = 2/9. Understanding this probability is useful in situations involving random selection, games, or quality control processes.

Calculate the probability that a randomly selected student from a class of 30 students, where 18 are girls and 12 are boys, is a boy.

In this scenario, we have a total of 30 students, which includes 18 girls and 12 boys. The event of selecting a boy has 12 favorable outcomes. The probability can be calculated using P(E) = Number of favorable outcomes / Total number of possible outcomes. Thus, P(Boy) = 12/30 = 2/5. This probability is a practical way to analyze gender representation in groups, which can be relevant for demographic studies or event planning.

If the probability of winning a game is 0.45, what is the probability of losing the game?

To find the probability of losing the game, we recognize that an event can either occur or not occur. The probability of losing is the complement of winning. If the probability of winning is given as P(Win) = 0.45, then the probability of losing can be calculated as P(Lose) = 1 - P(Win). Therefore, P(Lose) = 1 - 0.45 = 0.55. This approach of using complementary probabilities is essential in decision-making and risk assessment.

Probability - Mastery Worksheet

This worksheet challenges you with deeper, multi-concept long-answer questions from Probability to prepare for higher-weightage questions in Class 10.

Mastery

Questions

A box contains 3 red, 2 white, and 5 blue balls. If one ball is drawn at random, find the probability of drawing a red ball or a blue ball. Explain the steps and reasoning involved.

Total balls = 3 + 2 + 5 = 10. Probability of red = 3/10, Probability of blue = 5/10. P(red or blue) = P(red) + P(blue) = 3/10 + 5/10 = 8/10 = 4/5.

In a class of 30 students, 18 are girls and 12 are boys. If a student is selected at random, what is the probability that the student is either a girl or a boy? Justify your answer.

P(girl) = 18/30, P(boy) = 12/30. Since all students are either girls or boys, P(girl or boy) = 1.

Two dice are thrown simultaneously. Calculate the probability that the sum of the numbers on the dice is 8. Provide a detailed explanation.

Outcomes for sum 8: (2,6), (3,5), (4,4), (5,3), (6,2). Total outcomes = 6 × 6 = 36. P(sum = 8) = 5/36.

You roll a fair six-sided die. What is the probability of rolling a number less than 4? Show your calculations.

Outcomes less than 4: 1, 2, 3. Total outcomes = 6. P(number < 4) = 3/6 = 1/2.

A card is drawn from a standard deck of 52 cards. Determine the probability that it is not a face card. Explain how you arrived at your answer.

Face cards: 3 (Jack, Queen, King) of each suit = 12 in total. Non-face cards = 52 - 12 = 40. P(not face card) = 40/52 = 10/13.

A bag contains 4 yellow, 3 green, and 5 orange candies. If one candy is drawn, what is the probability that it is either yellow or green?

P(yellow) = 4/12, P(green) = 3/12. P(yellow or green) = 4/12 + 3/12 = 7/12.

In a lottery, the probability of winning a prize is 0.1. If a player enters the lottery 5 times, what is the probability that they win at least once?

P(no win in 5 draws) = (1 - 0.1)^5 = 0.9^5 ≈ 0.59049. Therefore, P(at least one win) = 1 - P(no win) ≈ 1 - 0.59049 ≈ 0.40951.

There are 50 apples in a basket, of which 15 are rotten. If you take out an apple at random, what is the probability that it is good? Show your working.

Good apples = 50 - 15 = 35. P(good apple) = 35/50 = 7/10.

A box contains 2 defective bulbs and 8 non-defective ones. If a bulb is drawn at random, what is the probability that it is defective? Provide the steps to calculate.

Total bulbs = 10. P(defective) = 2/10 = 1/5.

You flip three coins. What is the probability of getting at least one tail? Provide a thorough explanation.

Total outcomes = 2^3 = 8. Outcomes with no tails = (H,H,H). Thus, P(at least one tail) = 1 - P(no tails) = 1 - 1/8 = 7/8.

Probability - Challenge Worksheet

The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Probability in Class 10.

Challenge

Questions

Evaluate the implications of conditional probability in analyzing the risk of an event versus its complement in real-world crisis management scenarios.

Discuss the importance of understanding conditional probabilities in fields such as healthcare, emergency response, and finance. Provide examples where this knowledge can either mitigate risk or amplify it.

Analyze how the concept of expected value differs from actual probability and its implications in the field of gambling.

Explore how understanding the expected value can influence decision-making in gambling games. Include examples from various games to illustrate how expected values can deceive players.

Critically assess the misconceptions surrounding equally likely outcomes in everyday situations such as weather predictions or sports events.

Evaluate common beliefs about randomness and probability. Discuss how real-world factors complicate the assumption of equally likely outcomes.

Explore how the principles of probability are applied in sports analytics, focusing on player performance and game outcomes.

Discuss the methods used in sports analytics to predict game outcomes based on player statistics and historical data. Evaluate the risks of over-reliance on statistical models.

Investigate the ethical implications of using probabilities in decision-making processes in healthcare, particularly in treatment options.

Discuss the balance between statistical probability and individual patient care. Evaluate case studies where probability influenced treatment choices.

Debate the notion of certainty versus probability in legal decisions, particularly in the context of 'beyond a reasonable doubt' standard.

Analyze how probabilistic evidence is used in court and the implications of interpreting evidence in a legal context.

Assess the impact of probability in the stock market and the concept of risk versus reward.

Evaluate stock trading strategies that involve probabilities and expected returns. Discuss how market behaviors contradict theoretical models.

Explore the role of randomness in genetic mutations and its implications for evolutionary biology.

Discuss how randomness contributes to genetic diversity. Evaluate the potential consequences of certain mutations on species survival.

Evaluate the mathematical models of probability in weather forecasting and their effectiveness over time.

Discuss how advances in technology have improved probability estimates in meteorology. Analyze instances where forecasts have failed.

Analyze the use of probability in risk management for natural disasters, focusing on prediction and preparedness.

Explore how probabilities inform planning and response strategies for natural disasters like earthquakes or floods.

Probability Formula Sheet

Use this Class 10 Mathematics Probability Formula Sheet for quick revision before school exams and CBSE exams. It brings together the important formulas, key concepts, and worked examples in one place so students can revise faster and download a printable PDF for offline study.

Important Formulas

P(E) = \frac{Number \ of \ outcomes \ favourable \ to \ E}{Total \ number \ of \ possible \ outcomes}

P(E) defines the theoretical probability of an event E occurring. Represents ratio of favorable outcomes to total possible outcomes in an experiment.

P(not E) = 1 - P(E)

Calculates the probability of the complement of event E. Ensures that the sum of probabilities of an event and its complement equals 1.

P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2)

This is the addition rule for probabilities of two events E1 and E2. Useful for calculating the probability of either event occurring.

P(E_1 \cap E_2) = P(E_1) \times P(E_2) \ (if \ E_1, E_2 \ are \ independent)

Defines the probability of both events E1 and E2 occurring together, applicable when events are independent.

P(E) + P(not E) = 1

This identity reflects that the total probability for all outcomes in an experiment equals one, reinforcing the concept of complementary events.

P(E_1 \cup E_2 \cup E_3) = P(E_1) + P(E_2) + P(E_3) - P(E_1 \cap E_2) - P(E_1 \cap E_3) - P(E_2 \cap E_3) + P(E_1 \cap E_2 \cap E_3)

This is an extension for three events and is useful when finding the cumulative probability of multiple events.

P(E) = 0 \ (impossible \ event)

Indicates situations where an event cannot occur, such as drawing a number greater than 6 from a die.

P(E) = 1 \ (certain \ event)

Represents a situation that will definitely occur, e.g., rolling a number less than 7 on a die.

If \ n \ is \ the \ total \ number \ of \ marbles, \ P(red) = \frac{n_{red}}{n}

For a bag of different colored marbles, this formula calculates the probability of drawing a red marble.

Outcomes \ of \ a \ die:\ {1, 2, 3, 4, 5, 6}

Sets the foundation for calculating probabilities related to any event regarding the die's throw.

Worked Examples

P(E) = \frac{number \ of \ favourable \ outcomes}{total \ number \ of \ outcomes}

Fundamental equation for calculating any event's probability. Example: Tossing a coin to get heads.

P(Head) = \frac{1}{2}

The probability of getting heads when a fair coin is tossed once.

P(Tail) = \frac{1}{2}

The probability of getting tails when a fair coin is tossed once.

P(Yellow \ ball) = \frac{1}{3}

The probability of drawing a yellow ball from a bag containing red, blue, and yellow balls.

P(\text{sum}=8 \ in \ two \ dice) = \frac{5}{36}

Probability of obtaining a sum of 8 when rolling two six-sided dice.

P(\text{sum}>12) = 0

No outcomes yield a sum greater than 12 when throwing two dice.

P(\text{sum}\leq12) = 1

All possible outcomes of two dice rolls result in a sum of 12 or less.

P(Boy) = \frac{1}{2}

The probability of a newborn being a boy, assuming equal likelihood for boys and girls.

P(Girl) = \frac{1}{2}

The probability of a newborn being a girl, assuming equal likelihood for boys and girls.

From \ n = 6, \ P(rolling \ a \ 4) = \frac{1}{6}

Finding the probability of rolling a 4 on a standard die with 6 faces.

Explore More Probability Resources

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Probability Frequently Asked Questions

Explore the Probability chapter in Class 10 Mathematics, covering theoretical concepts, equally likely outcomes, and applications with detailed explanations and examples to enhance understanding.

QWhat is probability in mathematics?

Probability in mathematics is the measure of the likelihood that an event will occur. It quantifies uncertainty and ranges from 0 (impossible event) to 1 (certain event). In this chapter, probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes when all outcomes are equally likely.

QWhat are equally likely outcomes?

Equally likely outcomes are those outcomes that have the same chance of occurring. For example, when tossing a fair coin, both heads and tails are equally likely outcomes, each having a probability of 1/2. This concept forms the basis for many probability calculations in this chapter.

QWhat is empirical probability?

Empirical probability, also known as experimental probability, is determined based on experimental trials. It is calculated as the number of times an event occurs divided by the total number of trials. In situations where an experiment can be repeated, this type of probability provides insight into the likelihood of an event’s occurrence.

QHow is theoretical probability defined?

Theoretical probability is calculated based on the assumption that all possible outcomes are equally likely. It is defined as the number of favorable outcomes divided by the total number of possible outcomes. This helps in predicting the outcome of experiments without conducting them.

QHow do you calculate the probability of getting heads when tossing a coin?

To calculate the probability of getting heads when tossing a fair coin, you consider the total number of possible outcomes (which are heads and tails: 2). The number of favorable outcomes for getting heads is 1. Thus, the probability P(heads) = Number of favorable outcomes / Total number of outcomes = 1/2.

QWhat is an elementary event?

An elementary event is an event that consists of a single outcome. For example, in a coin toss, getting heads is an elementary event as it has only one outcome that occurs. Similarly, drawing a single card from a deck of cards, where obtaining an Ace is an elementary event as well.

QWhat is the complementary event?

The complementary event of an event E is defined as the event that E does not occur. If the probability of E occurring is P(E), then the probability of the complementary event occurring is given by P(not E) = 1 - P(E). This concept is crucial for solving probability problems.

QHow is the probability of drawing a red ball from a bag calculated?

To calculate the probability of drawing a red ball from a bag containing a mix of colored balls, count the number of red balls and divide it by the total number of balls. For example, if there are 4 red balls and 1 blue ball, the probability P(red) = Number of red balls / Total balls = 4/(4+1) = 4/5.

QCan the probability of an event be greater than 1?

No, the probability of any event cannot be greater than 1. Probability is a measure of the likelihood of an event occurring, and its value ranges from 0 to 1, where 0 represents an impossible event and 1 represents a certain event.

QWhat examples illustrate the concept of complementary events?

A common example of complementary events is when a die is rolled. If event A is rolling an even number (2, 4, or 6), the complementary event (not A) is rolling an odd number (1, 3, or 5). The sum of their probabilities equals 1, illustrating how complementary events function.

QWhat is the significance of the sum of probabilities of all outcomes?

The sum of the probabilities of all possible outcomes of an experiment must equal 1. This fundamental rule ensures that every possible outcome is accounted for, reflecting the certainty that one of the outcomes will occur when the experiment is performed.

QHow do you find the probability of a number appearing when rolling a die?

To find the probability of rolling a specific number on a fair die, identify that the die has 6 faces. The number of favorable outcomes for rolling one specific number (like a 3) is 1. Thus, the probability P(rolling a 3) = 1/6, as there are 6 equally likely outcomes.

QWhat is a sure event?

A sure event is an event that is certain to occur. For example, when drawing a card from a complete deck, obtaining a card that is either a heart, diamond, spade, or club is a sure event, with a probability of 1, as all cards belong to one of these suits.

QWhat does it mean if an event's probability is 0?

If an event's probability is 0, it means that the event cannot occur. For instance, the probability of rolling a 7 on a standard die is 0 since there are no outcomes that result in a 7, illustrating an impossible event.

QHow can we apply probability in real-life situations?

Probability can be applied in various real-life situations to predict outcomes, assess risks, and make informed decisions. For example, in finance, probability helps in assessing investments; in healthcare, it guides treatment decisions based on outcome likelihoods.

QWhat is the impact of sample size on empirical probability?

The sample size significantly impacts empirical probability. A larger sample size typically leads to a more accurate estimate of the probability of an event, as it reduces the impact of random variations and provides a clearer picture of the underlying distribution.

QWhat are some common mistakes in calculating probabilities?

Common mistakes in calculating probabilities include overlooking the total number of possible outcomes, neglecting to consider whether outcomes are equally likely, or miscalculating the number of favorable outcomes. Ensuring careful and systematic counting can help avoid these errors.

QHow to differentiate between independent and dependent events?

Independent events are those where the occurrence of one event does not affect the occurrence of the other (e.g., tossing a coin and rolling a die). In contrast, dependent events are those where one event affects the outcome of another (e.g., drawing a card without replacement).

QWhat is the law of total probability?

The law of total probability states that if you have a partition of the sample space, the total probability of an event can be found by summing the conditional probabilities of the event occurring given each part of the partition, multiplied by the probabilities of those parts.

QWhat role does probability play in statistics?

Probability plays a crucial role in statistics by providing a framework for making inferences about a population based on sample data. It allows statisticians to quantify uncertainty and assess the likelihood of events, contributing to hypothesis testing and decision-making.

QWhy is understanding probability important for students?

Understanding probability is essential for students as it enhances critical thinking and decision-making skills. It is applicable in various fields, from science and engineering to finance and social sciences, empowering students to analyze data, evaluate risks, and make informed choices.

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Probability PDF Downloads

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Probability Official Textbook PDF

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Probability Revision Guide

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Probability Formula Sheet

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Probability Practice Worksheet

Solve basic and application-based questions from Probability.

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Probability Mastery Worksheet

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Probability Challenge Worksheet

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Probability Question Bank

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Probability Flashcards

Revise key terms and definitions from Probability with interactive flashcards. Quick recall practice for CBSE Class 10 Mathematics.

These flash cards cover important concepts from Probability in Mathematics for Class 10 (Mathematics).

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What is probability?

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Probability is the measure of the likelihood that an event will occur, expressed as a number between 0 and 1.

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What does 'equally likely outcomes' mean?

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Equally likely outcomes are outcomes in an experiment that have the same chance of occurring.

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3/19

What is the formula for theoretical probability?

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P(E) = (Number of favorable outcomes) / (Total number of possible outcomes).

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What is empirical probability?

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Empirical probability is defined as P(E) = (Number of trials in which the event happened) / (Total number of trials).

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What is a sure event?

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A sure event is one that is certain to happen and has a probability of 1.

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What is an impossible event?

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An impossible event cannot happen and has a probability of 0.

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What are complementary events?

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Complementary events are two outcomes of an event that cannot occur at the same time. P(E) + P(not E) = 1.

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What's the probability of drawing a red ball from a bag with 4 red and 1 blue ball?

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P(red) = 4/5, since there are 4 favorable outcomes out of 5 total possible outcomes.

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What is the probability of getting heads when tossing a fair coin?

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P(head) = 1/2, as there are two equally likely outcomes: heads or tails.

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What are the outcomes when rolling a fair die?

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The outcomes are 1, 2, 3, 4, 5, and 6, each with an equal probability of 1/6.

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What is the probability of getting a sum of 8 when rolling two dice?

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The combinations to get 8 are (2,6), (3,5), (4,4), (5,3), and (6,2), hence P(sum=8) = 5/36.

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What is the probability that two people have the same birthday?

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If there are 365 days, the probability is P(same) = 1 - (364/365) = 1/365.

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What is the probability of selecting a girl from a class of 25 girls and 15 boys?

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P(girl) = 25/40 = 5/8.

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What's the probability of picking a white marble from a box with 3 blue, 2 white, and 4 red marbles?

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P(white) = 2/9.

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What is the probability of getting at least one head when tossing two coins?

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P(at least one head) = 3/4.

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What does it mean if P(E) + P(not E) = 1?

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This means that if one event occurs, the other cannot occur, representing certainty that one of them will happen.

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What is the probability of drawing an Ace from a standard deck of 52 cards?

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P(Ace) = 4/52 = 1/13.

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What are the possible values for probability?

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Probability values range from 0 to 1, inclusive.

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What is a common mistake in calculating probabilities?

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Assuming all outcomes are equally likely when they are not.

Practice Probability with Interactive Duels

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