Question

let a and b be independent events with p(a) = 0.6 and p(b) = 0.3. complete the sentence. the conditional probability of a given b is, and the conditional probability of b given a is.

Explanation:

Step1: Recall the definition of independent events and conditional probability

For two independent events \( A \) and \( B \), the conditional probability formula is \( P(A|B)=\frac{P(A\cap B)}{P(B)} \). And since \( A \) and \( B \) are independent, \( P(A\cap B) = P(A)\times P(B) \). But also, for independent events, the conditional probability \( P(A|B)=P(A) \) and \( P(B|A)=P(B) \) because the occurrence of one event does not affect the probability of the other.

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

Given \( P(A) = 0.6 \) and \( A \), \( B \) are independent, so \( P(A|B)=P(A)=0.6 \).

Step3: Find \( P(B|A) \)

Given \( P(B) = 0.3 \) and \( A \), \( B \) are independent, so \( P(B|A)=P(B)=0.3 \).

Answer:

The conditional probability of \( A \) given \( B \) is \( 0.6 \), and the conditional probability of \( B \) given \( A \) is \( 0.3 \).