Question

let a and b be two independent events. complete the paragraph. the conditional probability of a given b is the same as the probability of. the conditional probability of b given a is the same as the probability of.

Explanation:

Step1: Recall the formula for conditional probability

For independent events \(A\) and \(B\), \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Since \(A\) and \(B\) are independent, \(P(A\cap B) = P(A)\times P(B)\). Then \(P(A|B)=\frac{P(A)\times P(B)}{P(B)}=P(A)\).

Step2: Recall the formula for conditional probability

Also, \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). Because \(P(A\cap B) = P(A)\times P(B)\) (for independent events), \(P(B|A)=\frac{P(A)\times P(B)}{P(A)}=P(B)\).

Answer:

The conditional probability of \(A\) given \(B\) is the same as the probability of \(A\). The conditional probability of \(B\) given \(A\) is the same as the probability of \(B\).