Shaquille O’Neal is a legendary NBA player, renowned for being one of the most dominant forces in the history of the league, particularly due to his impact on the game and his prowess in the paint. However, he was also infamous for his low free throw percentage of 52.7% throughout his career. Opposing teams often resorted to intentionally fouling him to give him more free throw attempts, a strategy known as “Hack-a-Shaq.” For the following questions, suppose he attempted 10 free throws in a game. Q2 What is the probability that O’Neal made at least 7 free throws among 10 attempts?

To solve this problem, we can use the **binomial probability formula**, as each free throw attempt can be viewed as a Bernoulli trial (either a success or failure), with each attempt having an independent probability of success.

The formula for the probability of getting exactly \( k \) successes in \( n \) trials is:

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

where:
- \( n = 10 \) (the number of trials, or free throw attempts),
- \( p = 0.527 \) (Shaquille O'Neal's probability of making a free throw),
- \( k \) is the number of successful free throws.

We want to find the probability that he makes **at least 7 out of 10** free throws, which means calculating \( P(X \geq 7) \), where \( X \) is the number of successful free throws.

### Step 1: Calculate \( P(X \geq 7) \)

To find \( P(X \geq 7) \), we need to sum the probabilities from \( X = 7 \) to \( X = 10 \):

\[
P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
\]

### Step 2: Calculate Each Term Individually

1. **For \( P(X = 7) \):**
   \[
   P(X = 7) = \binom{10}{7} (0.527)^7 (1 - 0.527)^{10 - 7}
   \]

2. **For \( P(X = 8) \):**
   \[
   P(X = 8) = \binom{10}{8} (0.527)^8 (1 - 0.527)^{10 - 8}
   \]

3. **For \( P(X = 9) \):**
   \[
   P(X = 9) = \binom{10}{9} (0.527)^9 (1 - 0.527)^{10 - 9}
   \]

4. **For \( P(X = 10) \):**
   \[
   P(X = 10) = \binom{10}{10} (0.527)^{10} (1 - 0.527)^{10 - 10}
   \]

Let's calculate these values to find the probability.The probability that Shaquille O'Neal made at least 7 out of 10 free throws is approximately **0.220**, or **22.0%**.

Would you like more details on the binomial calculation process or further explanations on any part of this problem?

Here are some related questions you may find interesting:

1. What is the probability that Shaquille O'Neal makes exactly 5 free throws out of 10?
2. How does the probability change if his free throw percentage improves to 60%?
3. What is the probability that he makes at most 3 free throws in 10 attempts?
4. How would the probability change if he attempted 15 free throws instead of 10?
5. Can we approximate this probability using a normal distribution? If so, how accurate would it be?

**Tip:** In probability problems, visualizing with a probability distribution table or graph can often clarify patterns or likelihoods.

Shaquille O’Neal is a legendary NBA player, renowned for being one of the most dominant forces in the history of the league, particularly due to his impact on the game and his prowess in the paint. However, he was also infamous for his low free throw percentage of 52.7% throughout his career. Opposing teams often resorted to intentionally fouling him to give him more free throw attempts, a strategy known as 'Hack-a-Shaq.' For the following questions, suppose he attempted 10 free throws in a game. What is the probability that O’Neal made at least 7 free throws among 10 attempts?

Discover how to calculate the probability of Shaquille O'Neal making at least 7 free throws out of 10 attempts, given his career free throw percentage of 52.7%. This problem utilizes the binomial probability formula and covers key concepts in probability and statistics, making it suitable for Grades 10-12.

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