The probability the Rosa makes a free throw is 0.78

If she shoots 9 free throws what is the probability that she makes:

 

exactly 5 free throws?

This problem follows a **binomial distribution** since there are two possible outcomes for each shot: success (making a free throw) or failure (missing a free throw). 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 making exactly \( k \) free throws.
- \( n \) is the total number of trials (in this case, 9 free throws).
- \( k \) is the number of successes (in this case, 5 free throws).
- \( p \) is the probability of success on each trial (in this case, 0.78).
- \( \binom{n}{k} \) is the binomial coefficient, which can be calculated as:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

For this problem:
- \( n = 9 \)
- \( k = 5 \)
- \( p = 0.78 \)

Let’s calculate the probability.

First, calculate \( \binom{9}{5} \):

\[
\binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126
\]

Now, use the binomial formula:

\[
P(X = 5) = 126 \cdot (0.78)^5 \cdot (0.22)^4
\]

Let me compute that.The probability that Rosa makes exactly 5 free throws out of 9 is approximately **0.0852** or **8.52%**.

Would you like more details or have any other questions?

Here are 5 related questions to explore further:
1. What is the probability Rosa makes at least 5 free throws?
2. How would the probability change if she shot 12 free throws instead of 9?
3. What is the expected number of free throws she will make out of 9?
4. How does the probability change if Rosa's free throw success rate decreases to 0.6?
5. What is the probability Rosa makes fewer than 3 free throws?

**Tip:** The binomial distribution is useful when you are dealing with independent trials, where the outcome of one does not affect the others, such as repeated free throw attempts.

The probability that Rosa makes a free throw is 0.78. If she shoots 9 free throws, what is the probability that she makes exactly 5 free throws?

Calculate the probability that Rosa makes exactly 5 free throws out of 9 attempts, given her success rate is 0.78. This binomial probability problem introduces key concepts such as the binomial distribution formula and the binomial coefficient. Perfect for students in Grades 9-12 studying probability and statistics.

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