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 departed on time given that it arrives on time.
The probability that an airplane flight departs on time is 0.89. The probability that a flight arrives on time is 0.86 .
The probability that a flight departs and arrives on time is 0.82 .
The probability that a flight departed on time given that it arrives on time is (Round to the nearest thousandth as needed.)
Answer
So, the probability that a flight departed on time given that it arrives on time is \(\boxed{0.953}\).
Step 1 ：We are given the probability of event A (flight departs on time) as 0.89, event B (flight arrives on time) as 0.86 and event A and B (flight departs and arrives on time) as 0.82.
Step 2 ：We are asked to find the conditional probability of event A given that event B has occurred.
Step 3 ：We can use the formula for conditional probability to solve this problem: \(P(B \mid A)=\frac{P(A \text { and } B)}{P(A)}\).
Step 4 ：Substituting the given values into the formula, we get \(P(A \mid B) = \frac{0.82}{0.86}\).
Step 5 ：Solving this gives us a probability of 0.953.
Step 6 ：So, the probability that a flight departed on time given that it arrives on time is \(\boxed{0.953}\).
