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At LaGuardia Airport for a certain nightly flight, the probability that it will rain is 0.09 and the probability that the flight will be delayed is 0.07 . The probability that it will not rain and the flight will leave on time is 0.86 . What is the probability that the flight would be delayed when it is raining? Round your answer to the nearest thousandth.
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At LaGuardia Airport for a certain nightly flight, the probability that it will rain is $0$ . $09$ and the probability that the flight will be delayed is $0$ . $07$ . The probability that it will not rain and the flight will leave on time is $0$ . $86$ . What is the probability that the flight would be delayed when it is raining? Round your answer to the nearest thousandth. $\newline$ Answer:

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Find probabilities using the addition rule

Full solution

Q. At LaGuardia Airport for a certain nightly flight, the probability that it will rain is

0

.

09

and the probability that the flight will be delayed is

0

.

07

. The probability that it will not rain and the flight will leave on time is

0

.

86

. What is the probability that the flight would be delayed when it is raining? Round your answer to the nearest thousandth.

\newline

Answer:

Events Denotation: Let's denote the events as follows: $\newline$ R: It will rain. $\newline$ D: The flight will be delayed. $\newline$ \(\newlineeg R\): It will not rain. $\newline$ \(\newlineeg D\): The flight will leave on time. $\newline$ We are given the following probabilities: $\newline$ $P(R) = 0.09$ $\newline$ $P(D) = 0.07$ $\newline$ P(\(\newlineeg R \text{ and } $\newline$ eg D) = 0.86\) $\newline$ We need to find the probability that the flight would be delayed when it is raining, which is $P(D|R)$ . $\newline$ First, we need to find $P(R \text{ and } D)$ , the probability that it will rain and the flight will be delayed. We can use the Complement Rule and the Addition Rule of Probability to find this. $\newline$ The Complement Rule states that P(A) = 1 - P(\(\newlineeg A)\), and the Addition Rule states that $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ . $\newline$ We can find $P(R \text{ or } D)$ by using the Complement Rule on \(\newlineeg D\) $0$ : $\newline$ \(\newlineeg D\) $1$
Find $P(R \text{ and } D)$ : Now, we can use the Addition Rule to find $P(R \text{ and } D)$ : $P(R \text{ and } D) = P(R) + P(D) - P(R \text{ or } D) = 0.09 + 0.07 - 0.14$
Calculate $P(R \text{ and } D)$ : Let's calculate $P(R \text{ and } D)$ : $P(R \text{ and } D) = 0.09 + 0.07 - 0.14 = 0.02$
Find $P(D|R)$ : Finally, we can find $P(D|R)$ , the probability that the flight would be delayed when it is raining, by using the definition of conditional probability: $\newline$ $P(D|R) = \frac{P(R \text{ and } D)}{P(R)}$
Calculate $P(D|R)$ : Let's calculate $P(D|R)$ : $P(D|R) = \frac{0.02}{0.09}$
Calculate $P(D|R)$ : Let's calculate $P(D|R)$ : $P(D|R) = \frac{0.02}{0.09}$ After performing the division, we get: $P(D|R) \approx 0.222$

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