Sure! Let's tackle this problem step by step.
1. Understanding the Problem:
- Katie is a 75% free throw shooter, which means the probability of her making a single free throw is 0.75.
- She takes 15 free throws, so we are dealing with a binomial distribution where the number of trials (n) is 15, and the probability of success (p) is 0.75.
- We need to find the probability that she makes exactly 12 free throws out of the 15, denoted as [tex]\( P(X = 12) \)[/tex], where [tex]\( X \)[/tex] is the number of successful free throws.
2. Binomial Probability Formula:
The probability of getting exactly [tex]\( k \)[/tex] successes in [tex]\( n \)[/tex] independent Bernoulli trials is given by the binomial probability formula:
[tex]\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\][/tex]
where:
- [tex]\( \binom{n}{k} \)[/tex] is the binomial coefficient, calculated as [tex]\( \frac{n!}{k!(n-k)!} \)[/tex].
- [tex]\( n \)[/tex] is the total number of trials (15 in this case).
- [tex]\( k \)[/tex] is the number of successful trials (12 in this case).
- [tex]\( p \)[/tex] is the probability of a single success (0.75).
- [tex]\( 1-p \)[/tex] is the probability of a single failure (0.25).
3. Substituting Values:
Plugging in the values:
- [tex]\( n = 15 \)[/tex]
- [tex]\( k = 12 \)[/tex]
- [tex]\( p = 0.75 \)[/tex]
- [tex]\( 1-p = 0.25 \)[/tex]
Thus, the formula becomes:
[tex]\[
P(X = 12) = \binom{15}{12} (0.75)^{12} (0.25)^{3}
\][/tex]
4. Calculating
Through the calculation of the above formula, we find that:
[tex]\[
P(X = 12) \approx 0.22519906517118218
\][/tex]
5. Matching with Given Choices:
The given answer choices are:
- 0.764
- 0.236
- 0.225
The calculated probability 0.22519906517118218 is closest to 0.225.
Therefore, the correct answer is:
[tex]\[
\boxed{0.225}
\][/tex]
