Question

let a and b be two events with the following probabilities.

• the probability that a occurs is 0.6.
• the probability that b occurs is 0.9.
• the probability that b occurs given that a occurs is 0.3.

what is the conditional probability that a occurs given that b occurs?
write your answer as a decimal. do not round.

Explanation:

Step1: Use the formula for conditional probability

The formula for conditional probability is \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). We can find \(P(A\cap B)\) from this formula. Given \(P(A) = 0.6\) and \(P(B|A)=0.3\), we have \(P(A\cap B)=P(B|A)\times P(A)\).

$$P(A\cap B)=0.3\times0.6 = 0.18$$

Step2: Use Bayes' theorem

Bayes' theorem states that \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). We know \(P(A\cap B) = 0.18\) and \(P(B)=0.9\).

$$P(A|B)=\frac{0.18}{0.9}$$

$$P(A|B)=0.2$$

Answer:

\(0.2\)