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To solve this problem, we can model the situation using the **binomial distribution** because we are dealing with a fixed number of trials (7 at-bats), and the outcome for each trial (hit or no hit) is binary.

The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

Where:
- \( n = 7 \) (the number of at-bats),
- \( k = 7 \) (the number of successful hits),
- \( p = 0.275 \) (the probability of a hit in each at-bat),
- \( \binom{n}{k} \) is the binomial coefficient.

### Step 1: Compute the binomial coefficient
The binomial coefficient \( \binom{7}{7} \) is simply:

\[
\binom{7}{7} = 1
\]

### Step 2: Compute the probability
Now substitute the values into the formula:

\[
P(X = 7) = 1 \times (0.275)^7 \times (1 - 0.275)^{7 - 7}
\]

Simplifying:

\[
P(X = 7) = (0.275)^7 \times (0.725)^0
\]

Since \( (0.725)^0 = 1 \), we are left with:

\[
P(X = 7) = (0.275)^7
\]

### Step 3: Calculate the result
Now calculate \( (0.275)^7 \):

\[
(0.275)^7 \approx 2.24 \times 10^{-5}
\]

So, the probability that the player gets exactly 7 hits in 7 at-bats is approximately:

\[
P(X = 7) \approx 0.0000224
\]

Thus, the probability is approximately **0.0000224**, or **0.00224%** when converted to percentage.

Would you like further clarification on any part of the calculation?

Here are 5 related questions:
1. What is the probability of the player getting exactly 3 hits in 7 at-bats?
2. How would the probability change if the player's batting average increased to 0.300?
3. What is the expected number of hits the player would get in 7 at-bats?
4. How can we calculate the probability of the player getting at least 4 hits in 7 at-bats?
5. What is the variance of the number of hits in 7 at-bats?

**Tip:** The binomial distribution is useful when you are modeling situations with fixed numbers of trials and binary outcomes, such as success/failure or hit/no hit.

A baseball player has a batting average of 0.275. What is the probability that he has exactly 7 hits in his next 7 at-bats?

Calculate the probability of a baseball player with a 0.275 batting average getting exactly 7 hits in his next 7 at-bats using the binomial distribution. Learn the step-by-step process involving the binomial coefficient and probability calculations. Suitable for high school students.

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