This chapter explores the basic concepts and definitions of probability, highlighting its significance in predicting outcomes in uncertain situations.
Probability – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Probability chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
P(E) = Number of outcomes favourable to E / Number of all possible outcomes
P(E) represents the probability of event E. This formula calculates the likelihood of an event occurring by dividing the number of favourable outcomes by the total number of possible outcomes. Example: Probability of getting a head in a coin toss is 1/2.
P(E') = 1 - P(E)
P(E') is the probability of the complement of event E, meaning the event does not occur. It's derived by subtracting the probability of E from 1. Useful for finding the probability of an event not happening.
P(E or F) = P(E) + P(F) - P(E and F)
This formula calculates the probability of either event E or event F occurring. It accounts for overlapping events by subtracting the probability of both events occurring together.
P(E and F) = P(E) × P(F) for independent events
For independent events E and F, the probability of both occurring is the product of their individual probabilities. Example: Probability of getting two heads in two coin tosses is 1/2 × 1/2 = 1/4.
P(E) = 0 for an impossible event
An impossible event has no chance of occurring, hence its probability is 0. Example: Probability of getting a 7 in a single die throw is 0.
P(E) = 1 for a certain event
A certain event is guaranteed to occur, hence its probability is 1. Example: Probability of getting a number less than 7 in a die throw is 1.
Sum of probabilities of all elementary events = 1
The total probability of all possible outcomes in an experiment sums up to 1. This is a fundamental property of probability.
P(E) ≤ 1
The probability of any event E is always less than or equal to 1, ensuring that probabilities are within a valid range.
P(E) ≥ 0
The probability of any event E is always greater than or equal to 0, meaning negative probabilities are not possible.
P(E|F) = P(E and F) / P(F)
P(E|F) is the conditional probability of E given F. It calculates the probability of E occurring under the condition that F has already occurred.
Equations
Probability of drawing an ace from a deck of cards: P(Ace) = 4/52
There are 4 aces in a 52-card deck. The probability is calculated by dividing the number of aces by the total number of cards.
Probability of not drawing an ace: P(Not Ace) = 48/52
This is the complement of drawing an ace. Calculated by subtracting the probability of drawing an ace from 1 or directly as (52-4)/52.
Probability of getting a number >4 in a die throw: P(>4) = 2/6
Only 5 and 6 are greater than 4 in a die. Thus, the probability is 2 favourable outcomes out of 6 possible.
Probability of two friends having different birthdays: P(Different) = 364/365
Assuming 365 days in a year and ignoring leap years, the probability that two people have different birthdays is 364/365.
Probability of two friends having the same birthday: P(Same) = 1/365
The complement of having different birthdays. Directly calculated as 1/365 for any given pair.
Probability of drawing a red ball from a bag with 3 red and 5 black balls: P(Red) = 3/8
The probability is the ratio of red balls to the total number of balls.
Probability of not drawing a red ball: P(Not Red) = 5/8
This is the probability of drawing a black ball, calculated as the complement of drawing a red ball.
Probability of getting at least one head in two coin tosses: P(At least one H) = 3/4
The favourable outcomes are HT, TH, HH. Total outcomes are 4. Thus, 3/4.
Probability of sum of 8 in two dice throws: P(Sum=8) = 5/36
There are 5 combinations that sum to 8: (2,6), (3,5), (4,4), (5,3), (6,2). Total outcomes are 36.
Probability of sum ≤12 in two dice throws: P(Sum≤12) = 1
The maximum sum in two dice throws is 12, making this a certain event with probability 1.
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