To solve this problem, we can model the number of hits $H$ as a binomial random variable, since each at-bat can be considered a Bernoulli trial where a hit occurs with a certain probability.

Step-by-Step Solution:

1. Define the parameters:

   • Let $p$ be the probability of getting a hit (batting average). $p = 0.188$
   • Let $n$ be the number of at-bats in the season. $n = 295$

   $H \sim \text{Binomial}(n, p)$

2. Calculate the expected number of hits:

   • The mean $\mu$ of a binomial distribution is calculated as: $\mu = n \cdot p = 295 \cdot 0.188 \approx 55.46$
   • The variance $\sigma^2$ is: $\sigma^2 = n \cdot p \cdot (1 - p) = 295 \cdot 0.188 \cdot 0.812 \approx 45.08$
   • The standard deviation $\sigma$: $\sigma \approx \sqrt{45.08} \approx 6.71$

3. Use a normal approximation to the binomial distribution: Since $n$ is large (295), we can use the normal approximation to the binomial distribution: $H \approx N(\mu, \sigma^2) = N(55.46, 45.08)$

4. Apply the continuity correction: To approximate the probability $P(H \geq 34)$ using the normal distribution, we apply a continuity correction: $P(H \geq 34) \approx P\left(Z \geq \frac{34 - 0.5 - 55.46}{6.71}\right)$ $P(H \geq 34) \approx P\left(Z \geq \frac{-21.96}{6.71}\right)$ $P(H \geq 34) \approx P(Z \geq -3.27)$

5. Find the cumulative probability: Using standard normal distribution tables or a calculator: $P(Z \geq -3.27) \approx 0.9994$

Final Answer:

The probability that the player gets 34 or more hits in the season is approximately 0.9994.

Would you like further details or have any questions?

Related Questions:

1. How would the probability change if the player’s batting average increased to 0.200?
2. What is the expected number of hits for this player if they have 500 at-bats?
3. How does the continuity correction impact the approximation accuracy?
4. What would be the probability of the player getting fewer than 20 hits?
5. If the player had 50 at-bats, could we still use the normal approximation?

Tip:

The normal approximation is best used when $n$ is large and $p$ is not too close to 0 or 1. For small $n$ or extreme values of $p$, the binomial distribution should be used directly.