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In an experiment, the probability that event $A$ occurs is $\frac{2}{5}$ and the probability that event $B$ occurs is $\frac{2}{3}$ . 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{2}{5}

and the probability that event

B

occurs is

\frac{2}{3}

. If

A

and

B

are independent events, what is the probability that

A

and

B

both occur?

\newline

Simplify any fractions.

Question Prompt: question_prompt: What's the probability that both event $A$ and event $B$ happen if they're independent?
Calculation of $P(A \text{ and } B)$ : $P(A \text{ and } B) = P(A) \times P(B)$ cuz they're independent, right? So, we gotta multiply the probabilities of $A$ and $B$ .
Calculation of P(A): $P(A) = \frac{2}{5}$ and $P(B) = \frac{2}{3}$ . Let's do the math: $\frac{2}{5} \times \frac{2}{3}$ .
Calculation of P(B): Multiplying the numerators: $2 \times 2 = 4$ . Multiplying the denominators: $5 \times 3 = 15$ . So, $P(A \text{ and } B) = \frac{4}{15}$ .
Final Calculation: Now, we gotta check if $\frac{4}{15}$ can be simplified. But nah, $4$ and $15$ don't have common factors other than $1$ . So, we're done here.

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