At LaGuardia Airport for a certain nightly flight, the probability that it will rain is $0$ . $19$ and the probability that the flight will be delayed is $0$ . $17$ . The probability that it will rain and the flight will be delayed is $0$ . $05$ . What is the probability that it is not raining if the flight leaves on time? Round your answer to the nearest thousandth. $\newline$ Answer:

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Algebra 2

Find probabilities using the addition rule

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

0

19

and the probability that the flight will be delayed is

0

17

. The probability that it will rain and the flight will be delayed is

0

05

. What is the probability that it is not raining if the flight leaves on time? 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. $\newline$ We are given the following probabilities: $\newline$ $P(R) = 0.19$ $\newline$ $P(D) = 0.17$ $\newline$ $P(R \text{ and } D) = 0.05$ $\newline$ We want to find the probability that it is not raining given that the flight leaves on time. This can be expressed as $P(\text{Not } R | \text{Not } D)$ , which is the conditional probability of it not raining given that the flight is not delayed.
Find Probability Not Delayed: First, we need to find the probability of the flight not being delayed, which is $P(\text{Not } D)$ . This is the complement of the flight being delayed, so we calculate it as: $\newline$ $P(\text{Not } D) = 1 - P(D)$ $\newline$ $P(\text{Not } D) = 1 - 0.17$ $\newline$ $P(\text{Not } D) = 0.83$
Find Probability Not Rain and Not Delayed: Next, we need to find the probability of it not raining and the flight not being delayed, which is $P(\text{Not } R \text{ and Not } D)$ . This can be found by taking the complement of the probability of either raining or the flight being delayed. Using the Addition Rule of Probability, we have: $\newline$ $P(R \text{ or } D) = P(R) + P(D) - P(R \text{ and } D)$ $\newline$ $P(R \text{ or } D) = 0.19 + 0.17 - 0.05$ $\newline$ $P(R \text{ or } D) = 0.31$ $\newline$ Now, we find the complement of $P(R \text{ or } D)$ to get $P(\text{Not } R \text{ and Not } D)$ : $\newline$ $P(\text{Not } R \text{ and Not } D) = 1 - P(R \text{ or } D)$ $\newline$ $P(\text{Not } R \text{ and Not } D) = 1 - 0.31$ $\newline$ $P(\text{Not } R \text{ and Not } D) = 0.69$
Calculate Conditional Probability: Now we can use the definition of conditional probability to find $P(\text{Not } R | \text{Not } D)$ . The formula for conditional probability is: $\newline$ $P(\text{Not } R | \text{Not } D) = \frac{P(\text{Not } R \text{ and } \text{Not } D)}{P(\text{Not } D)}$ $\newline$ Substituting the values we have found: $\newline$ $P(\text{Not } R | \text{Not } D) = \frac{0.69}{0.83}$ $\newline$ $P(\text{Not } R | \text{Not } D) \approx 0.8313$
Final Probability Calculation: Finally, we round the answer to the nearest thousandth as requested: $P(\text{Not } R | \text{Not } D) \approx 0.831$