The probability of the student getting exactly 15 correct answers by guessing is approximately 0.0606, which corresponds to option a. 0.0606.
Binomial Probability: In this scenario, we have 48 independent trials (questions) with two possible outcomes (correct or incorrect) for each. The probability of getting a correct answer by guessing is 1/4 (as there's only one correct answer among 4 options). The probability of getting an incorrect answer is 3/4 (since 1 - 1/4 = 3/4).
We want the probability of getting exactly 15 correct answers out of 48. This can be calculated using the binomial probability formula:
P(k successes in n trials) = (n choose k) * p^k * (1-p)^(n-k)
where:
k is the number of successes (correct answers) = 15
n is the total number of trials (questions) = 48
p is the probability of success (getting a correct answer) = 1/4
Calculating this will give you a very large number, but it's not needed for the next step.
Normal Approximation:
For large n and moderate values of p (0.2 < p < 0.8), the binomial distribution can be approximated by the normal distribution. This allows for easier calculations using the normal probability density function (PDF).
To use the normal approximation, we need to calculate the following:
Mean (μ): n * p = 48 * 1/4 = 12
Standard deviation (σ): √(n * p * (1-p)) = √(48 * 1/4 * 3/4) = √(36) = 6
Finding the Probability:
We want the probability of getting exactly 15 correct answers, which corresponds to a specific range in the normal distribution. This range is centered around the mean (μ) with a width of one standard deviation (σ) on either side (12 ± 6, or 6 to 18).
Using the normal PDF with μ = 12 and σ = 6, we can calculate the area under the curve within this specific range. This area represents the probability of getting 15 correct answers (rounded to four decimal places):
P(6 <= X <= 18) ≈ 0.0606
Therefore, the probability of the student getting exactly 15 correct answers by guessing is approximately 0.0606, which corresponds to option a. 0.0606.