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.