Final answer:
To find the probability that Mickey Mantle would get exactly 5 hits in 15 times at bat, we can use the normal approximation or the binomial distribution. The normal approximation gives a probability of about 0.396, while the binomial distribution gives a probability of about 0.367. Although there is a slight difference between the two results, they are relatively close.
Step-by-step explanation:
To find the probability that Mickey Mantle would get exactly 5 hits in 15 times at bat, we can use the normal approximation. Since the batting average is 0.365, we can assume that each at bat is an independent trial with a success (getting a hit) probability of 0.365.
- Calculate the mean of the number of hits in 15 at bats: mean = 15 * 0.365 = 5.475.
- Calculate the standard deviation of the number of hits in 15 at bats: standard deviation = sqrt(15 * 0.365 * (1 - 0.365)) = 1.793.
- Convert the problem into a z-score by subtracting the mean and dividing by the standard deviation: z = (5 - 5.475) / 1.793 = -0.265.
- Use a standard normal distribution table or calculator to find the probability of getting a z-score of -0.265. The closest probability is 0.3959.
Using a binomial distribution, we can also find the probability. Using the formula for the probability of getting exactly x successes in n trials with a success probability of p, we have:
binomial probability = nCk * p^k * (1-p)^(n-k), where nCk is the number of combinations of n items taken k at a time.
For this problem, n = 15, k = 5, and p = 0.365. Plugging in the values:
binomial probability = 15C5 * 0.365^5 * (1-0.365)^(15-5)
binomial probability = 3003 * 0.365^5 * 0.635^10
binomial probability ≈ 0.367.
Comparing the results, we can see that the normal approximation gives a probability of about 0.396, while the binomial distribution gives a probability of about 0.367. Although there is a slight difference between the two results, they are relatively close.