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This season, the probability that the Yankees will win a game is $0.48$ and the probability that the Yankees will score $5$ or more runs in a game is $0.57$ . The probability that the Yankees lose and score fewer than $5$ runs is $0.31$ . What is the probability that the Yankees will lose when they score fewer than $5$ runs? Round your answer to the nearest thousandth.

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Find probabilities using the addition rule

Full solution

Q. This season, the probability that the Yankees will win a game is

0.48

and the probability that the Yankees will score

5

or more runs in a game is

0.57

. The probability that the Yankees lose and score fewer than

5

runs is

0.31

. What is the probability that the Yankees will lose when they score fewer than

5

runs? Round your answer to the nearest thousandth.

Events Denoted: Let's denote the events as follows: $\newline$ W: The Yankees win a game. $\newline$ S: The Yankees score $5$ or more runs in a game. $\newline$ L: The Yankees lose a game. $\newline$ F: The Yankees score fewer than $5$ runs in a game. $\newline$ We are given the following probabilities: $\newline$ $P(W) = 0.48$ (Probability that the Yankees win) $\newline$ $P(S) = 0.57$ (Probability that the Yankees score $5$ or more runs) $\newline$ $P(L \text{ and } F) = 0.31$ (Probability that the Yankees lose and score fewer than $5$ runs) $\newline$ We need to find the probability that the Yankees will lose when they score fewer than $5$ runs, which can be represented as $P(L | F)$ , the conditional probability of L given F. $\newline$ To find $P(L | F)$ , we need to know $P(F)$ , the probability that the Yankees score fewer than $5$ runs. Since the probability of scoring $5$ or more runs is $0.57$ , the probability of scoring fewer than $5$ runs is the complement of that event. $\newline$ $P(F) = 1 - P(S)$ $\newline$ $P(F) = 1 - 0.57$ $\newline$ $P(F) = 0.43$ $\newline$ Now we can calculate $P(L | F)$ using the formula for conditional probability: $\newline$ $P(S) = 0.57$ $1$ $\newline$ Substituting the given values, we get: $\newline$ $P(S) = 0.57$ $2$
Calculate $P(F)$ : Performing the division to find $P(L | F)$ : $P(L | F) = \frac{0.31}{0.43}$ $P(L | F) \approx 0.7209302325581395$ Rounding to the nearest thousandth, we get: $P(L | F) \approx 0.721$

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