Conditional probability Formula

The potential of an event or outcome occurring based on the existence of a preceding event or outcome is known as conditional probability. It's calculated by multiplying the likelihood of the previous occurrence by the probability of the next, or conditional, event.

The concepts of independent and dependent events are introduced here. Consider a student who goes on vacation twice a week, excluding Sunday. What are the possibilities that he will likewise take a leave on Saturday in the same week if he will be absent from school on Tuesday? These examples of probability are known as a conditional probability because the occurrence of one event has an effect on the occurrence of the next event.

Definition of Conditional probability Formula

Conditional probability is the chance of any event A occurring when another event B in relation to A has already occurred. P(A|B) illustrates it.

Because we need to calculate the odds of event A occurring, only a piece that is common to both A and B is sufficient to reflect the probability of A occurring when B has already occurred. The intersection of both events A and B, i.e. A B, represents the common component of the events.

Formula

When two events intersect, the formula for conditional probability for the occurrence of two events is as follows:

P(A/B) = N(A ∩ B) / N(B)

P(B/A) = N(A ∩ B) / N(P(A/B) = N(A ∩ B) / N(B)

Where P(A/B) is the likelihood of A occurring after B has occurred.

The number of components shared by both A and B is N(A B).

The number of items in B is N(B), which cannot be zero.

Total number of elements in the sample space is denoted by N.

P(A/B) = N(A ∩ B) / N(B)

Because N(A B)/N and N(B)/N denote the ratio of the number of favourable outcomes to the total number of outcomes, the probability is indicated.

Thus, N(A ∩ B)/N can be written as P(A ∩ B) and N(B)/N as P(B).

p(A/B) = p(A ∩ B) / P(B)

Therefore, P( A ∩ B) = P(B) P(A|B) if P(B) ≠ 0

Also, the probability of occurrence of B when A has already occurred is given by,

P(B|A) = P(B ∩ A)/P(A)

Conditional Probability and Bayes Theorem

The likelihood of an event occurring in response to any situation is defined by Bayes' theorem. It's taken into account in the case of conditional probability. This is also known as the formula for determining the likelihood of "causes."

P(A/B) = P(B / A)P(A)/ P(B)

Properties

Let E and F be events of a sample space S of an experiment, then we have: P(S|F) = P(F|F) = 1

P(S|F) = P(F|F) = 1.

If A and B are any two events of a sample space S and F is an event of S such that P(F) ≠ 0, then; P((A ∩ B)|F) = P(A|F) + P(B|F) - P((A ∩ B)|F)

P((A ∪ B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F)

P(A′|B) = 1 − P(A|B)

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Conditional probability Formula