A multiple choice exam has 20 questions; there are four choices for each question.
A student guesses the answer to every question. Find the chance that he guesses correctly between four and seven times.
Find the minimum score the instructor can set so that the probability that a student will pass just by guessing is 20% or less.
thank you
To find the chance that the student guesses correctly between four and seven times, we need to determine the probability of getting exactly four, five, six, or seven correct answers out of 20.
The probability of guessing a single question correctly is 1/4, since there are four choices for each question. The probability of guessing a single question incorrectly is 3/4.
To find the probability of getting exactly four correct answers out of 20, we use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- C(n, k) is the number of ways to choose k successes from n trials,
- p is the probability of a single success (in this case, 1/4),
- (1-p) is the probability of a single failure (in this case, 3/4),
- n is the total number of trials (in this case, 20),
- k is the number of successes we want (in this case, 4).
Using this formula, we can calculate the probabilities for getting exactly four, five, six, and seven correct answers, and then sum them up to get the total probability of guessing correctly between four and seven times.
P(4 correct answers) = C(20, 4) * (1/4)^4 * (3/4)^(20-4)
P(5 correct answers) = C(20, 5) * (1/4)^5 * (3/4)^(20-5)
P(6 correct answers) = C(20, 6) * (1/4)^6 * (3/4)^(20-6)
P(7 correct answers) = C(20, 7) * (1/4)^7 * (3/4)^(20-7)
Finally, we can add these probabilities to find the total probability:
Total probability = P(4 correct answers) + P(5 correct answers) + P(6 correct answers) + P(7 correct answers)
To find the minimum score the instructor can set so that the probability of passing just by guessing is 20% or less, we need to find the number of correct answers a student needs to pass. Since there are 20 questions and each question has four choices, the passing score is determined by the expected score of guessing. The probability of guessing a single question correctly is 1/4, so the expected score is:
Expected score = (1/4) * 20 = 5
To find the minimum passing score, we need to find the number of correct answers a student needs to have a probability of 20% or less of passing. We can subtract the probability of guessing correctly between four and seven times from 1 (since the total probability of passing is 1), and set it to 20%:
1 - Total probability = 20%
Then, we can solve for the minimum passing score:
Number of correct answers = passing score - expected score
Now let's calculate the probabilities and the minimum passing score using these formulas.