Conditional probability P(A | B) indicates the probability of even ‘A’ happening given that the even B happened.

We can easily understand above formula using below diagram. Since B has already happened, the sample space reduces to B. So the probability of A happening becomes P(A ∩ B) divided by P(B)

**Example:**

In a batch, there are 80% C programmers, and 40% are Java and C programmers. What is the probability that a C programmer is also Java programmer?

Let A --> Event that a student is Java programmer B --> Event that a student is C programmer P(A|B) = P(A ∩ B) / P(B) = (0.4) / (0.8) = 0.5 So there are 50% chances that student that knows C also knows Java

**Product Rule:**

Derived from above definition of conditional probability by multiplying both sides with P(B)

P(A ∩ B) = P(B) * P(A|B)

**Exercise:**

**1)** What is the value of P(A|A)?

**2)** Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = 1/2, the values of P(A | B) and P(B | A) respectively are (GATE CS 2003)

(A) 1/4, 1/2

(B) 1/2, 1/14

(C) 1/2, 1

(D) 1, 1/2

See this for solution.

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