This chapter introduces the fundamental concepts of probability, including conditional probability and its applications which are essential for understanding uncertainty in random experiments.
Probability – Formula & Equation Sheet
Essential formulas and equations from Mathematics Part - II, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Probability chapter of Mathematics Part - II. 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|F) = P(E ∩ F) / P(F) for P(F) ≠ 0
P(E|F) denotes the conditional probability of event E given that event F has occurred. It quantifies how the occurrence of F influences the likelihood of E.
P(E ∩ F) = P(E) * P(F|E)
This formula describes the joint probability of events E and F occurring together, calculated as the probability of E and the probability of F given E.
P(E ∪ F) = P(E) + P(F) - P(E ∩ F)
This formula calculates the probability of either event E or F occurring, ensuring that both events are not double-counted.
P(E') = 1 - P(E)
P(E') represents the probability of the complement event of E, meaning E does not occur. Useful for simplifying calculations.
P(E|F) + P(E'|F) = 1
This property reflects that the total probability for all outcomes must sum to 1, showing that if F occurs, either E must occur or not occur.
P(A ∪ B | F) = P(A | F) + P(B | F) - P(A ∩ B | F)
This formula extends the addition rule to conditional probabilities, allowing the calculation of the probability of either A or B given F.
P(A ∩ B) = P(A) * P(B|A)
It denotes the multiplication rule for the joint occurrence of events A and B, where A influences the occurrence of B.
If E and F are independent, P(E ∩ F) = P(E) * P(F)
This defines the condition for independence between two events, where the occurrence of one does not impact the probability of the other.
P(A | B) = P(A) when A and B are independent
When events A and B are independent, the occurrence of B does not change the probability of A.
Total Probability: P(A) = Σ P(E_i) * P(A|E_i)
This theorem is used to compute the total probability of event A based on partition events E_i, ensuring comprehensive coverage of all possibilities.
Equations
P(E|F) = P(E ∩ F) / P(F)
Defines conditional probability of E given F.
P(E') = 1 - P(E)
Probability of the complement of event E.
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Union of events formula.
P(E|F) + P(E'|F) = 1
Sum of probabilities of event E and its complement given F.
P(E ∩ F) = P(E) * P(F|E)
Joint probability using conditional probability.
P(A ∩ B) = P(A) * P(B) if A and B are independent
Product of probabilities for independent events.
P(E|F) + P(E'|F) = 1
Total probability of all outcomes given F.
P(A ∪ B | F) = P(A | F) + P(B | F) - P(A ∩ B | F)
Conditional addition rule.
P(A) = Σ P(E_i) * P(A|E_i)
Total probability theorem.
P(E ∩ F) = P(E) * P(F|E) = P(F) * P(E|F)
Multiplication rule for joint events.
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