Question

by rewriting the formula for the multiplication rule, you can write a formula for finding conditional probabilities. the conditional probability of event b occurring, given that event a has occurred, is \\(p(b \mid a) = \frac{p(a \text{ and } b)}{p(a)}\\). use the information below to find the probability that a flight arrives on time given that it departed on time.

the probability that an airplane flight departs on time is 0.92.
the probability that a flight arrives on time is 0.86.
the probability that a flight departs and arrives on time is 0.84.

the probability that a flight arrives on time given that it departed on time is .
(round to the nearest thousandth as needed.)

Explanation:

Identify the given probabilities

Using the Conditional Probability Notation knowledge point
Let \(D\) be the event that a flight departs on time, and \(A\) be the event that a flight arrives on time.

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Define the target conditional probability

We want to find the probability that a flight arrives on time given that it departed on time. This is represented as:

$$ P(A \mid D) $$

Apply the conditional probability formula

Using the Conditional Probability Notation knowledge point

$$ P(A \mid D) = \frac{P(D \text{ and } A)}{P(D)} $$

Substitute the values and calculate

$$ P(A \mid D) = \frac{0.84}{0.92} \approx 0.913043 $$

Round to the nearest thousandth

Rounding \(0.913043\) to three decimal places yields:

$$ P(A \mid D) \approx 0.913 $$

Answer:

The probability that a flight arrives on time given that it departed on time is <blank>0.913</blank>.