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In an experiment, the probability that event $A$ occurs is $\frac{2}{7}$ , the probability that event $B$ occurs is $\frac{6}{7}$ , and the probability that events $A$ and $B$ both occur is $\frac{2}{9}$ . $\newline$ What is the probability that $A$ occurs given that $B$ occurs? $\newline$ Simplify any fractions. $\newline$ ____ $\newline$

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Find conditional probabilities

Full solution

Q. In an experiment, the probability that event

A

occurs is

\frac{2}{7}

, the probability that event

B

occurs is

\frac{6}{7}

, and the probability that events

A

and

B

both occur is

\frac{2}{9}

.

\newline

What is the probability that

A

occurs given that

B

occurs?

\newline

Simplify any fractions.

\newline

____

\newline

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 $P(A \text{ and } B) = \frac{2}{9}$ and $P(B) = \frac{6}{7}$ . So, $P(A|B) = \frac{\frac{2}{9}}{\frac{6}{7}}$ .
Multiply Fractions: To divide the fractions, we multiply by the reciprocal of the second fraction: $(\frac{2}{9}) \times (\frac{7}{6})$ .
Simplify Fraction: Now, multiply the numerators and denominators: $(2 \times 7) / (9 \times 6) = 14 / 54$ .
Simplify Fraction: Now, multiply the numerators and denominators: $(2 \times 7) / (9 \times 6) = 14 / 54$ .Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is $2$ : $14 / 54 = 7 / 27$ .

More problems from Find conditional probabilities

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college students running for student government class president. The candidates include

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In an experiment, the probability that event

A

\frac{2}{7}

, the probability that event

B

\frac{7}{9}

, and the probability that events

A

B

\frac{1}{9}

\newline

A

B

independent events?

\newline

\newline

\text{yes}

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At a science museum, visitors can compete to see who has a faster reaction time. Competitors watch a red screen, and the moment they see it turn from red to green, they push a button. The machine records their reaction times and also asks competitors to report their gender.

\newline

The probability that a competitor reacted in over

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0.6

, the probability that a competitor was female is

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, and the probability that a competitor reacted in over

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seconds and was female is

0.3

\newline

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Write your answer as a whole number, decimal, or simplified fraction.

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Question

A restaurant server claims that \(28\)% of the time he leaves candy with the bill, the customer tips well.

\newline

If the server's claim is true, and one day he leaves candy with \(4\) customers' bills, what is the probability that exactly \(3\) of those customers will tip well?

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Question

There are two different raffles you can enter.

\newline

In raffle A, one ticket will win a `$720` prize, and the other tickets will win nothing. There are

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\newline

Raffle B is for a `$370` prize. Out of

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\newline

Which raffle is a better deal?

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\newline

\text{[A]Raffle A}

\newline

\text{[B]Raffle B}

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Question

X

is a normally distributed random variable with mean

49

and standard deviation

25

\newline

What is the probability that

X

24

74

?

\newline

0.68-0.95-0.997

rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

\newline

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X

is a normally distributed random variable with mean

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and standard deviation

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\newline

What is the probability that

X

62

68

?

\newline

Write your answer as a decimal rounded to the nearest thousandth.

\newline

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Question

Complete the statement. Round your answer to the nearest thousandth.

\newline

In a population that is normally distributed with mean

43

and standard deviation

3

30\%

of the values are those less than ______.

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