Questions: 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 A) = P(A 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 0.87. The probability that a flight departs and arrives on time is 0.81. The probability that a flight arrives on time given that it departed on time is (Round to the nearest thousandth as needed.)

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 A) = P(A 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 0.87. The probability that a flight departs and arrives on time is 0.81.

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

Transcript text: 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 0.87 . The probability that a flight departs and arrives on time is 0.81 . The probability that a flight arrives on time given that it departed on time is (Round to the nearest thousandth as needed.)

Solution

Solution Steps

Step 1: Define the Given Probabilities

We are given the following probabilities:

$ P(A) = P(\text{depart on time}) = 0.92 $
$ P(B) = P(\text{arrive on time}) = 0.87 $
$ P(A \text{ and } B) = P(\text{depart and arrive on time}) = 0.81 $

Step 2: Apply the Conditional Probability Formula

To find the conditional probability $ P(B \mid A) $, we use the formula: \[ P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)} \] Substituting the known values: \[ P(B \mid A) = \frac{0.81}{0.92} \]

Step 3: Calculate the Conditional Probability

Calculating the above expression: \[ P(B \mid A) \approx 0.8804347826086957 \] Rounding this value to the nearest thousandth gives: \[ P(B \mid A) \approx 0.8804 \]