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 0.1. The probability that it will not rain and the flight will leave on time is 0.8. What is the probability that it is raining and the flight is delayed? Round your answer to the nearest thousandth.

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Denote Events: Let's denote the events as follows: R: It will rain. D: The flight will be delayed. N: It will not rain and the flight will leave on time.  We are given the following probabilities: P(R) = 0.12 P(D) = 0.1 P(N) = 0.8  We need to find the probability that it is raining and the flight is delayed, which is P(R and D).Find P(R or D): First, we need to find the probability that it will rain or the flight will be delayed, which is P(R or D). This can be found using the complement of the probability that it will not rain and the flight will leave on time, which is P(N).  Since P(N) is the probability that it will not rain and the flight will leave on time, the complement is the probability that it will rain or the flight will be delayed, which is 1 - P(N).  So, P(R or D) = 1 - P(N).Calculate P(R or D): Now, let's calculate P(R or D) using the value of P(N) given:  P(R or D) = 1 - P(N) P(R or D) = 1 - 0.8 P(R or D) = 0.2Use Inclusion-Exclusion Principle: Next, we can use the Inclusion-Exclusion Principle to find P(R and D). The principle states that for any two events, R and D:  P(R or D) = P(R) + P(D) - P(R and D)  We can rearrange this formula to solve for P(R and D):  P(R and D) = P(R) + P(D) - P(R or D)Substitute Values: Now, let's substitute the values we have into the equation:  P(R and D) = P(R) + P(D) - P(R or D) P(R and D) = 0.12 + 0.1 - 0.2Calculate P(R and D): Calculating the value of P(R and D):  P(R and D) = 0.12 + 0.1 - 0.2 P(R and D) = 0.22 - 0.2 P(R and D) = 0.02Round the Answer: Finally, we round the answer to the nearest thousandth as requested:  P(R and D) ≈ 0.020

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