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In an experiment, the probability that event $A$ occurs is $\frac{3}{7}$ , the probability that event $B$ occurs is $\frac{1}{2}$ , and the probability that events $A$ and $B$ both occur is $\frac{3}{8}$ . What is the probability that $A$ occurs given that $B$ occurs? Simplify any fractions.

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Find conditional probabilities

Full solution

Q. In an experiment, the probability that event

A

occurs is

\frac{3}{7}

, the probability that event

B

occurs is

\frac{1}{2}

, and the probability that events

A

and

B

both occur is

\frac{3}{8}

. What is the probability that

A

occurs given that

B

occurs? Simplify any fractions.

Use Conditional Probability Formula: To find the probability that $A$ occurs given that $B$ occurs, we use the formula for conditional probability: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$ .
Calculate $P(A|B)$ : We know $P(A \text{ and } B) = \frac{3}{8}$ and $P(B) = \frac{1}{2}$ .
Multiply Fractions: Now, let's do the calculation: $P(A|B) = \frac{3}{8} / \frac{1}{2}.$
Simplify Multiplication: To divide by a fraction, we multiply by its reciprocal. So, $P(A|B) = \frac{3}{8} \times \frac{2}{1}.$
Reduce Fraction: Multiplying the fractions, we get $P(A|B) = \frac{3 \times 2}{8 \times 1}$ .
Final Probability: Simplifying the multiplication, $P(A|B) = \frac{6}{8}$ .
Final Probability: Simplifying the multiplication, $P(A|B) = \frac{6}{8}$ .We can reduce the fraction $\frac{6}{8}$ by dividing both numerator and denominator by $2$ .
Final Probability: Simplifying the multiplication, $P(A|B) = \frac{6}{8}$ .We can reduce the fraction $\frac{6}{8}$ by dividing both numerator and denominator by $2$ .After simplifying, we get $P(A|B) = \frac{3}{4}$ .

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