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In an experiment, the probability that event $A$ occurs is $\frac{1}{2}$ and the probability that event $B$ occurs is $\frac{4}{9}$ . If $A$ and $B$ are independent events, what is the probability that $A$ and $B$ both occur? $\newline$ Simplify any fractions.

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Independence and conditional probability

Full solution

Q. In an experiment, the probability that event

A

occurs is

\frac{1}{2}

and the probability that event

B

occurs is

\frac{4}{9}

. If

A

and

B

are independent events, what is the probability that

A

and

B

both occur?

\newline

Simplify any fractions.

Identify Independence: $P(A \text{ and } B) = P(A) \times P(B)$ cuz they're independent, so we gotta multiply the probabilities of $A$ and $B$ .
Calculate Individual Probabilities: $P(A) = \frac{1}{2}$ and $P(B) = \frac{4}{9}$ , so $P(A \text{ and } B) = \frac{1}{2} \times \frac{4}{9}$ .
Multiply Probabilities: Now we do the math: $\frac{1}{2} \times \frac{4}{9} = \frac{4}{18}$ .
Simplify Fraction: We can simplify $\frac{4}{18}$ by dividing both the numerator and the denominator by $2$ .
Final Result: After simplifying, we get $\frac{2}{9}$ .

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