Question

Mrs. Gruber gave her history class a multiple- choice quiz containing five questions. A student must answer at least four questions correctly to pass.

Greg decided to guess on every question. If each of the four possible answers to each ques- tion is equally likely to be chosen, what is the probability that Greg passed the quiz?

Answer 1

To calculate the probability that Greg passes the quiz, we must consider the probabilities of him getting exactly four questions right or all five. This is done using the binomial probability formula and then summing up the probabilities of both scenarios.

Step-by-step explanation:

The question asks about the probability that Greg passes a history class quiz by guessing on every question. This is a problem that involves understanding and calculating binomial probabilities, since for each question there are two possible outcomes: Greg guesses correctly, or he does not.

To calculate the probability of Greg passing, we need to consider two scenarios where he can pass: either he gets exactly four questions right, or he gets all five right. Since there are four options per question, the probability of guessing any single question correctly is 1/4, and hence guessing it incorrectly is 3/4. To pass the quiz, Greg needs to guess at least four questions correctly, which can happen in the following ways:

1. He answers four questions correctly and one question incorrectly.
2. He answers all five questions correctly.

To find the probabilities of these events, one must use the binomial probability formula:

P(X = k) = (n choose k) × (probability of success)^k × (probability of failure)^(n-k)

where 'n' represents the total number of questions, 'k' represents the number of correct answers needed, (n choose k) is the number of ways k successes can occur in n trials, and P(X = k) is the probability of having exactly k correct answers (= successes).

Let's calculate the probability for both scenarios:

1. Probability of getting exactly four correct: (5 choose 4) × (1/4)^4 × (3/4)^1
2. Probability of getting all five correct: (5 choose 5) × (1/4)^5 × (3/4)^0

We then add these two probabilities together to find the total probability that Greg passes his quiz.

It's important to note that the student's question and our calculations are unrelated to actual historical knowledge, making this purely a mathematical problem of calculating probabilities.