This season, the probability that the Yankees will win a game is 0.51 and the probability that the Yankees will score 5 or more runs in a game is 0.57 . The probability that the Yankees win and score 5 or more runs is 0.45 . What is the probability that the Yankees would score 5 or more runs when they lose the game? Round your answer to the nearest thousandth. Answer:

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Define Events and Probabilities: Define the events and given probabilities. Let's denote the events as follows: W: The Yankees win a game. R: The Yankees score 5 or more runs in a game. We are given the following probabilities: P(W) = 0.51 (Probability that the Yankees win) P(R) = 0.57 (Probability that the Yankees score 5 or more runs) P(W and R) = 0.45 (Probability that the Yankees win and score 5 or more runs)Calculate Probability of Yankees Losing: Calculate the probability that the Yankees lose a game. Since the probability of winning is P(W) = 0.51, the probability of losing, which we'll call P(L), is the complement of P(W). P(L) = 1 - P(W) P(L) = 1 - 0.51 P(L) = 0.49Use Conditional Probability Definition: Use the definition of conditional probability. We want to find the probability that the Yankees score 5 or more runs given that they lose the game, which is P(R|L). According to the definition of conditional probability: P(R|L) = P(R and L) / P(L) We already have P(L), but we need to find P(R and L), which is the probability that the Yankees score 5 or more runs and lose the game.Find P(R and L) Using Complement: Find P(R and L) using the complement of the event (W and R). The event (R and L) is the complement of (W and R) within the context of R. This means that: P(R and L) = P(R) - P(W and R) Substituting the given values, we get: P(R and L) = 0.57 - 0.45 P(R and L) = 0.12Calculate P(R|L): Calculate P(R|L) using the values from steps 2 and 4. Now we can substitute P(R and L) and P(L) into the conditional probability formula: P(R|L) = P(R and L) / P(L) P(R|L) = 0.12 / 0.49 P(R|L) ≈ 0.245Round to Nearest Thousandth: Round the answer to the nearest thousandth. P(R|L) ≈ 0.245 When rounded to the nearest thousandth, this is: P(R|L) ≈ 0.245

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