A high school baseball player has a 0.221 batting average. In one game, he gets 9 at bats. What is the probability he will get at least 4 hits in the game?


This problem is a classic example of a **binomial probability** situation. The player either gets a hit or doesn't for each at-bat, and the probability of getting a hit is given by his batting average. Let's break down the problem step by step:

### Problem Parameters:
- The probability of success (getting a hit) in one at-bat: \( p = 0.221 \)
- The probability of failure (not getting a hit) in one at-bat: \( q = 1 - p = 1 - 0.221 = 0.779 \)
- The number of at-bats: \( n = 9 \)
- We are looking for the probability that he gets **at least 4 hits**, which means \( X \geq 4 \), where \( X \) is the number of hits.

### Step 1: Binomial Distribution Formula
The binomial probability formula is:

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

Where:
- \( P(X = k) \) is the probability of getting exactly \( k \) hits.
- \( \binom{n}{k} \) is the binomial coefficient, which counts the number of ways to choose \( k \) successes (hits) out of \( n \) trials (at-bats).
- \( p^k \) is the probability of getting \( k \) hits.
- \( (1-p)^{n-k} \) is the probability of the remaining \( n-k \) at-bats being failures.

### Step 2: Probability of Getting At Least 4 Hits
To find \( P(X \geq 4) \), we sum the probabilities of getting 4, 5, 6, 7, 8, and 9 hits:

\[
P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9)
\]

It will be easier to compute the complement: \( P(X < 4) \), which is the probability of getting 0, 1, 2, or 3 hits, and subtract it from 1.

\[
P(X \geq 4) = 1 - P(X < 4)
\]

Thus, we calculate:
\[
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]

### Step 3: Calculation
I'll now compute these probabilities using the binomial formula.

Let me calculate these values for you.The probability that the player will get at least 4 hits in the game is approximately **0.116**, or about **11.6%**.

Would you like more details or have any questions about this result?

Here are 5 related questions for further exploration:
1. What is the probability of the player getting exactly 5 hits in the game?
2. How does the probability change if the player's batting average improves to 0.300?
3. What is the probability of getting no hits in 9 at-bats?
4. Can you calculate the expected number of hits in a game with 9 at-bats?
5. How would the probability change if the player had 12 at-bats instead of 9?

**Tip:** In binomial problems, it's often easier to calculate the complement (the opposite of what you're looking for) and subtract it from 1.

A high school baseball player has a 0.221 batting average. In one game, he gets 9 at bats. What is the probability he will get at least 4 hits in the game?

Discover how to calculate the probability of a high school baseball player with a 0.221 batting average getting at least 4 hits in 9 at-bats. This step-by-step solution covers the binomial probability theorem and uses the complement rule for easier computation. Perfect for high school students (Grades 9-12).

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