Question

If the probability that a randomly selected student from a college is good at mathematics is 0.6, then the probability of having exactly two students who are good at mathematics in a group of 8 students standing in front of the college is:

• A. \( \frac{2^6 \times 3^2 \times 7}{5^8} \)
• B. \( \frac{2^6 \times 3^2 \times 7}{5^6} \)
• C. \( \frac{2^8 \times 3^2 \times 7}{5^6} \)
• D. \( \frac{2^8 \times 3^2 \times 7}{5^8} \)

Hint:

Use the binomial distribution for probability calculations involving a fixed number of successes in independent trials.

Answer 1

The Correct Option is D

We are tasked with finding the probability of having exactly two students who are good at mathematics in a group of 8 students, given that the probability of a student being good at mathematics is \(0.6\). Step 1: Identify the probability distribution This is a binomial probability problem. The probability of exactly \(k\) successes (students good at mathematics) in \(n\) trials (students) is given by: \[ P(k) = \binom{n}{k} p^k (1-p)^{n-k} \] where:
\(n = 8\) (number of students),
\(k = 2\) (number of students good at mathematics),
\(p = 0.6\) (probability of a student being good at mathematics),
\(1-p = 0.4\) (probability of a student not being good at mathematics).
Step 2: Compute the binomial coefficient The binomial coefficient \(\binom{8}{2}\) represents the number of ways to choose 2 students out of 8. It is calculated as: \[ \binom{8}{2} = \frac{8!}{2! \cdot 6!} = \frac{8 \times 7}{2 \times 1} = 28 \] Step 3: Compute the probability Substitute the values into the binomial probability formula: \[ P(2) = \binom{8}{2} \cdot (0.6)^2 \cdot (0.4)^6 \] Simplify the terms: - \((0.6)^2 = 0.36\), - \((0.4)^6 = \left(\frac{2}{5}\right)^6 = \frac{2^6}{5^6}\). Thus: \[ P(2) = 28 \cdot 0.36 \cdot \frac{2^6}{5^6} \] Simplify \(0.36\) as a fraction: - \(0.36 = \frac{36}{100} = \frac{9}{25} = \frac{3^2}{5^2}\). Now substitute: \[ P(2) = 28 \cdot \frac{3^2}{5^2} \cdot \frac{2^6}{5^6} \] Combine the terms: \[ P(2) = 28 \cdot \frac{3^2 \cdot 2^6}{5^8} \] Express \(28\) as \(2^2 \times 7\): \[ P(2) = \frac{2^2 \times 7 \times 3^2 \times 2^6}{5^8} \] Combine the powers of \(2\): \[ P(2) = \frac{2^{2+6} \times 3^2 \times 7}{5^8} = \frac{2^8 \times 3^2 \times 7}{5^8} \] Step 4: Match with the options The probability matches option (4): \[ \frac{2^8 \times 3^2 \times 7}{5^8} \] Final Answer: \(\boxed{4}\)