A baseball player has a lifetime batting average of 0.188. If, in a season, this player has 295 "at bats", what is the probability he gets 34 or more hits?
Let H denote the number of hits.

Find the probability that the player gets 34 or more hits.

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.

A baseball player has a lifetime batting average of 0.188. If, in a season, this player has 295 'at bats', what is the probability he gets 34 or more hits? Let H denote the number of hits. Find the probability that the player gets 34 or more hits.

Explore the probability of a baseball player with a batting average of 0.188 achieving 34 or more hits in 295 at-bats. This problem demonstrates key statistical concepts, including the binomial distribution, normal approximation, and continuity correction. Suitable for undergraduate students studying probability and statistics.

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