Question

in an experiment, the probability that event a occurs is \\( \frac { 4 } { 9 } \\), the probability that event b occurs is \\( \frac { 3 } { 7 } \\), and the probability that events a and b both occur is \\( \frac { 2 } { 5 } \\).
what is the probability that a occurs given that b occurs?
simplify any fractions.

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Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\).

Step2: Substitute the given values into the formula

We are given that \(P(A\cap B)=\frac{2}{5}\) and \(P(B)=\frac{3}{7}\).
Substituting these values into the formula, we get \(P(A|B)=\frac{\frac{2}{5}}{\frac{3}{7}}\).

Step3: Simplify the fraction

To divide by a fraction, we multiply by its reciprocal. So \(P(A|B)=\frac{2}{5}\times\frac{7}{3}=\frac{14}{15}\).

Answer:

\(\frac{14}{15}\)