Answer :
Answer:
(
4
3
)
15
=(
4
3
)
15
=
4
15
3
15
Step-by-step explanation:
To find the probability that the student answers exactly 5 questions correctly, we'll use the binomial probability formula:
\[ P(X = k) = \binom{n}{k} \times p^k \times (1 - p)^{n - k} \]
Where:
- \( n \) is the number of trials (number of questions),
- \( k \) is the number of successes (number of questions answered correctly),
- \( p \) is the probability of success on each trial (probability of answering a question correctly), and
- \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose \( k \) successes out of \( n \) trials.
In this case:
- \( n = 20 \) (20 questions),
- \( k = 5 \) (5 questions answered correctly),
- \( p = \frac{1}{4} \) (probability of guessing a question correctly), and
- \( 1 - p = \frac{3}{4} \) (probability of guessing incorrectly).
Let's calculate:
\[ P(X = 5) = \binom{20}{5} \times \left(\frac{1}{4}\right)^5 \times \left(\frac{3}{4}\right)^{20 - 5} \]
\[ P(X = 5) = \binom{20}{5} \times \left(\frac{1}{4}\right)^5 \times \left(\frac{3}{4}\right)^{15} \]
Now, let's compute each part:
\[ \binom{20}{5} = \frac{20!}{5!(20 - 5)!} = \frac{20!}{5!15!} = \frac{20 \times 19 \times 18 \times 17 \times 16}{5 \times 4 \times 3 \times 2 \times 1} \]
\[ \left(\frac{1}{4}\right)^5 = \left(\frac{1}{4}\right)^5 = \frac{1}{4^5} = \frac{1}{1024} \]
\[ \left(\frac{3}{4}\right)^{15} = \left(\frac{3}{4}\right)^{15} = \frac{3^{15}}{4^{15}} \]
Now, let's calculate these values and multiply them together to find \( P(X = 5) \).
