Question

let a and b be independent events with ( p(a)=0.2 ) and ( p(b)=0.5 ). which of the following statements must be true?
the probability that a and b both occur is 0.2.
the probability that a and b both occur is 0.5.
the conditional probability that a occurs given that b occurs is 0.2.
the conditional probability that a occurs given that b occurs is 0.5.

Explanation:

Step1: Calculate \(P(A\cap B)\)

Since \(A\) and \(B\) are independent events, \(P(A\cap B)=P(A)\times P(B)\).
Given \(P(A) = 0.2\) and \(P(B)=0.5\), then \(P(A\cap B)=0.2\times0.5 = 0.1\). So the first two statements (about \(P(A\cap B)\) being \(0.2\) or \(0.5\)) are false.

Step2: Calculate \(P(A|B)\)

By the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\).
Since \(A\) and \(B\) are independent, \(P(A\cap B) = P(A)\times P(B)\), then \(P(A|B)=\frac{P(A)\times P(B)}{P(B)}=P(A)\).
Given \(P(A) = 0.2\), so \(P(A|B)=0.2\)

Answer:

The conditional probability that \(A\) occurs given that \(B\) occurs is \(0.2\).