In an experiment, the probability that event A occurs is 2/5, the probability that event B occurs is 5/6, and the probability that events A and B both occur is 1/3. What is the probability that A occurs given that B occurs? Simplify any fractions. ____

Use Formula: To find the probability of A given B, we use the formula P(A|B) = P(A and B) / P(B).Calculate Probabilities: We know P(A and B) = 1/3 and P(B) = 5/6.Divide Fractions: Now we calculate P(A|B) = (1/3) / (5/6).Simplify Result: To divide fractions, we multiply by the reciprocal of the second fraction: (1/3) * (6/5).Multiplying the numerators: 1 * 6 = 6. Multiplying the denominators: 3 * 5 = 15.So, P(A|B) = 6/15.We can simplify 6/15 by dividing both numerator and denominator by their greatest common divisor, which is 3.After simplifying, we get P(A|B) = 2/5.

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In an experiment, the probability that event $A$ occurs is $\frac{2}{5}$ , the probability that event $B$ occurs is $\frac{5}{6}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{3}$ . What is the probability that $A$ occurs given that $B$ occurs? Simplify any fractions.

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Precalculus

Find conditional probabilities

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Q. In an experiment, the probability that event

A

occurs is

\frac{2}{5}

, the probability that event

B

occurs is

\frac{5}{6}

, and the probability that events

A

and

B

both occur is

\frac{1}{3}

. What is the probability that

A

occurs given that

B

occurs? Simplify any fractions.

Use Formula: To find the probability of A given B, we use the formula $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$ .
Calculate Probabilities: We know $P(A \text{ and } B) = \frac{1}{3}$ and $P(B) = \frac{5}{6}$ .
Divide Fractions: Now we calculate $P(A|B) = \frac{1}{3} / \frac{5}{6}$ .
Simplify Result: To divide fractions, we multiply by the reciprocal of the second fraction: $(\frac{1}{3}) \times (\frac{6}{5})$ .
Simplify Result: To divide fractions, we multiply by the reciprocal of the second fraction: $(\frac{1}{3}) \times (\frac{6}{5})$ .Multiplying the numerators: $1 \times 6 = 6$ . $\newline$ Multiplying the denominators: $3 \times 5 = 15$ .
Simplify Result: To divide fractions, we multiply by the reciprocal of the second fraction: $(\frac{1}{3}) \times (\frac{6}{5})$ .Multiplying the numerators: $1 \times 6 = 6$ . $\newline$ Multiplying the denominators: $3 \times 5 = 15$ .So, $P(A|B) = \frac{6}{15}$ .
Simplify Result: To divide fractions, we multiply by the reciprocal of the second fraction: $(\frac{1}{3}) \times (\frac{6}{5})$ .Multiplying the numerators: $1 \times 6 = 6$ . Multiplying the denominators: $3 \times 5 = 15$ .So, $P(A|B) = \frac{6}{15}$ .We can simplify $\frac{6}{15}$ by dividing both numerator and denominator by their greatest common divisor, which is $3$ .
Simplify Result: To divide fractions, we multiply by the reciprocal of the second fraction: $(\frac{1}{3}) \times (\frac{6}{5})$ . Multiplying the numerators: $1 \times 6 = 6$ . Multiplying the denominators: $3 \times 5 = 15$ . So, $P(A|B) = \frac{6}{15}$ . We can simplify $\frac{6}{15}$ by dividing both numerator and denominator by their greatest common divisor, which is $3$ . After simplifying, we get $P(A|B) = \frac{2}{5}$ .