At LaGuardia Airport for a certain nightly flight, the probability that it will rain is 0.12 and the probability that the flight will be delayed is 0.15 . The probability that it will not rain and the flight will leave on time is 0.83 . What is the probability that the flight would leave on time when it is raining? Round your answer to the nearest thousandth. Answer:

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Events Denotation: Let's denote the events as follows: R: It will rain. D: The flight will be delayed. T: The flight will leave on time. We are given the following probabilities: P(R) = 0.12 P(D) = 0.15 P(not R and T) = 0.83 We need to find the probability that the flight will leave on time given that it is raining, which is P(T|R).Find Probability of Flight Leaving on Time: First, we need to find the probability that the flight will leave on time, which is P(T). We can use the complement rule since we know P(not R and T), which is the probability that it will not rain and the flight will leave on time. P(T) = P(not R and T) + P(R and T) However, we don't have P(R and T) directly, so we need to find it using the information we have.Find Probability of Flight Delayed: We can find P(R and T) by using the fact that P(R and T) is the complement of P(R and D), the probability that it will rain and the flight will be delayed. P(R and T) = P(R) - P(R and D) We don't have P(R and D) directly, but we can find it using the probability of the flight being delayed, P(D), and the probability that the flight will leave on time when it's not raining, P(not R and T). P(D) = P(R and D) + P(not R and D)Find Probability of Flight Leaving on Time Given Rain: We can rearrange the above equation to find P(R and D): P(R and D) = P(D) - P(not R and D) We know P(D) = 0.15, but we don't have P(not R and D) directly. However, we can find it using the complement of P(not R and T). P(not R and D) = P(not R) - P(not R and T)Find Probability of Flight Leaving on Time: Now we need to find P(not R), which is the complement of P(R). P(not R) = 1 - P(R) P(not R) = 1 - 0.12 P(not R) = 0.88Using the value of P(not R), we can find P(not R and D): P(not R and D) = P(not R) - P(not R and T) P(not R and D) = 0.88 - 0.83 P(not R and D) = 0.05Now we can find P(R and D): P(R and D) = P(D) - P(not R and D) P(R and D) = 0.15 - 0.05 P(R and D) = 0.10With P(R and D) found, we can now find P(R and T): P(R and T) = P(R) - P(R and D) P(R and T) = 0.12 - 0.10 P(R and T) = 0.02Now we can find P(T), the probability that the flight will leave on time: P(T) = P(not R and T) + P(R and T) P(T) = 0.83 + 0.02 P(T) = 0.85Finally, we can find the probability that the flight will leave on time given that it is raining, which is P(T|R): P(T|R) = P(R and T) / P(R) P(T|R) = 0.02 / 0.12 P(T|R) = 1/6 P(T|R) ≈ 0.167 (rounded to the nearest thousandth)

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