Final answer:
To calculate the probability that the student guesses more than 75 percent of the questions correctly, we can use the binomial distribution.
With three possible choices for each question and success defined as guessing correctly, we can find the probability of getting 4 or more correct answers out of 32 using the binomial probability formula. The probability is approximately 0.8022, or 80.22%.
Step-by-step explanation:
To find the probability that the student guesses more than 75 percent of the questions correctly, we can use the binomial distribution. Let's define success as guessing a question correctly, and failure as guessing incorrectly. The probability of success, p, is 1/3, since there are three possible choices and only one correct option. The probability of failure, q, is 1 - p, which is 2/3. Now, let's calculate the probability of getting 4 or more correct answers out of 32:
P(X ≥ 4) = 1 - P(X ≤ 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]
Using the binomial probability formula, we can calculate each individual term:
P(X = 0) = binompdf(32, 1/3, 0) = 0.6829
P(X = 1) = binompdf(32, 1/3, 1) = 0.3294
P(X = 2) = binompdf(32, 1/3, 2) = 0.1367
P(X = 3) = binompdf(32, 1/3, 3) = 0.0488
Now, let's substitute these values back into the equation to find the probability of getting 4 or more correct answers:
P(X ≥ 4) = 1 - [0.6829 + 0.3294 + 0.1367 + 0.0488] = 0.8022
Therefore, the probability that the student guesses more than 75 percent of the questions correctly is approximately 0.8022, or 80.22%.