Answer:
To find the probability that a baseball player with a lifetime batting average of 0.303 gets 125 or more hits in a season with 365 at-bats, we can use the binomial probability formula. In this case, it's a binomial distribution because each at-bat is a separate, independent trial with a probability of success (getting a hit) of 0.303.
The formula for the probability of getting exactly k successes in n trials, each with a probability p of success, is:
\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k} \]
Where:
- \( n \) is the number of trials (365 at-bats).
- \( k \) is the number of successes (hits, in this case, 125 or more).
- \( p \) is the probability of success in a single trial (lifetime batting average, 0.303).
First, let's calculate the probability of getting exactly 125 hits:
\[ P(X = 125) = \binom{365}{125} \cdot (0.303)^{125} \cdot (1 - 0.303)^{365 - 125} \]
Now, we need to calculate the probabilities for getting 126, 127, and so on, up to 365 hits, and sum them up because we want the probability of getting 125 or more hits:
\[ P(X \geq 125) = P(X = 125) + P(X = 126) + \ldots + P(X = 365) \]
Calculating this sum can be quite laborious by hand, but you can use statistical software or a calculator to simplify the process.
Using a calculator or statistical software, you can find the cumulative probability \( P(X \geq 125) \) for this binomial distribution. The result will give you the probability that the player gets 125 or more hits in the season.