Suppose a baseball pitcher throws fastballs 80 % of the time and curveballs 20 % of the time. Suppose a batter hits a home run on 8 % of all fastball pitches, and on 5 % of all curveball pitches. What is the probability that this batter will hit a home run on this pitcher's next pitch?
0.025
To find the probability that the batter will hit a home run on the pitcher's next pitch, we need to consider the probabilities of two separate events:
1. The probability that the next pitch will be a fastball, denoted as P(F).
2. The probability that the batter will hit a home run given that the next pitch is a fastball, denoted as P(H|F).
3. The probability that the next pitch will be a curveball, denoted as P(C).
4. The probability that the batter will hit a home run given that the next pitch is a curveball, denoted as P(H|C).
From the given information, we know that the pitcher throws fastballs 80% of the time (0.80) and curveballs 20% of the time (0.20). We also know that the batter hits a home run on fastballs 8% of the time (0.08) and on curveballs 5% of the time (0.05).
Now let's use this information to calculate the probability that the batter will hit a home run on the next pitch:
P(H) = P(H|F) * P(F) + P(H|C) * P(C)
P(H) = (0.08 * 0.80) + (0.05 * 0.20)
P(H) = 0.064 + 0.01
P(H) = 0.074
Therefore, the probability that the batter will hit a home run on the pitcher's next pitch is 0.074, or 7.4%.