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This season, the probability that the Yankees will win a game is 0.54 and the probability that the Yankees will score 5 or more runs in a game is 0.46. The probability that the Yankees lose and score fewer than 5 runs is 0.37. What is the probability that the Yankees win and score 5 or more runs? Round your answer to the nearest thousandth.

Question

This season, the probability that the Yankees will win a game is 0.54 and the probability that the Yankees will score 5 or more runs in a game is 0.46. The probability that the Yankees lose and score fewer than 5 runs is 0.37. What is the probability that the Yankees win and score 5 or more runs? Round your answer to the nearest thousandth.

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Solution

Let's denote the following events:

  • W W : The event that the Yankees win a game.
  • S S : The event that the Yankees score 5 or more runs in a game.

We are given:

  • P(W)=0.54 P(W) = 0.54
  • P(S)=0.46 P(S) = 0.46
  • P(Wc∩Sc)=0.37 P(W^c \cap S^c) = 0.37 , where Wc W^c is the event that the Yankees lose, and Sc S^c is the event that the Yankees score fewer than 5 runs.

We need to find P(W∩S) P(W \cap S) , the probability that the Yankees win and score 5 or more runs.

First, we use the fact that the total probability must sum to 1: P(W∩S)+P(W∩Sc)+P(Wc∩S)+P(Wc∩Sc)=1 P(W \cap S) + P(W \cap S^c) + P(W^c \cap S) + P(W^c \cap S^c) = 1

We know:

  • P(Wc∩Sc)=0.37 P(W^c \cap S^c) = 0.37
  • P(W)=0.54 P(W) = 0.54
  • P(S)=0.46 P(S) = 0.46

Next, we find P(Wc) P(W^c) and P(Sc) P(S^c) : P(Wc)=1−P(W)=1−0.54=0.46 P(W^c) = 1 - P(W) = 1 - 0.54 = 0.46 P(Sc)=1−P(S)=1−0.46=0.54 P(S^c) = 1 - P(S) = 1 - 0.46 = 0.54

Now, we use the law of total probability for P(Sc) P(S^c) : P(Sc)=P(W∩Sc)+P(Wc∩Sc) P(S^c) = P(W \cap S^c) + P(W^c \cap S^c) 0.54=P(W∩Sc)+0.37 0.54 = P(W \cap S^c) + 0.37 P(W∩Sc)=0.54−0.37=0.17 P(W \cap S^c) = 0.54 - 0.37 = 0.17

Similarly, we use the law of total probability for P(W) P(W) : P(W)=P(W∩S)+P(W∩Sc) P(W) = P(W \cap S) + P(W \cap S^c) 0.54=P(W∩S)+0.17 0.54 = P(W \cap S) + 0.17 P(W∩S)=0.54−0.17=0.37 P(W \cap S) = 0.54 - 0.17 = 0.37

Therefore, the probability that the Yankees win and score 5 or more runs is: P(W∩S)=0.37 P(W \cap S) = 0.37

Rounded to the nearest thousandth, the answer is: 0.370 0.370

This problem has been solved

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