A test is composed of six multiple choice questions where each question has 4 choices. If the answer choices for each question are equally likely, find the probability of answering 3 OR 4 questions correctly.

Please type your subject in the School Subject box. Any other words, including obscure abbreviations, are likely to delay responses from a teacher who knows that subject well.

R --- right

W --- wrong

so we want the case of
RRRWWW or RRRRWW and its possible arrangements

P(3 R) = C(6,3)(1/4^3(3/4)^3 = .131836
P(4 R) = C(6,4)(1/4)^4(3/4)^2 = .032959

P(3R or 4R) = the sum of those two
= .1648

To find the probability of answering 3 or 4 questions correctly, we need to consider two scenarios: when exactly 3 questions are answered correctly, and when exactly 4 questions are answered correctly. We will calculate the probability for each scenario and add them together to get the final probability.

Let's start with the scenario of answering exactly 3 questions correctly:
To answer 3 questions correctly, we need to choose 3 out of the 6 questions correctly and 3 out of the 4 choices correctly for each of those 3 questions. So, the probability of answering 3 questions correctly is calculated as follows:

Probability of answering 3 questions correctly = (number of ways to choose 3 questions correctly) * (probability of choosing the correct answer for each of those 3 questions)^3 * (probability of choosing the incorrect answer for the remaining 3 questions)^(6-3)

The number of ways to choose 3 questions correctly out of 6 questions is represented as "6 choose 3" and can be calculated using the binomial coefficient formula:

(6 choose 3) = 6! / (3! * (6-3)!) = 20

The probability of choosing the correct answer for each of those 3 questions is 1/4, and the probability of choosing the incorrect answer for the remaining 3 questions is 3/4. Substituting these values, we get:

Probability of answering 3 questions correctly = 20 * (1/4)^3 * (3/4)^3

Similarly, we calculate the probability of answering exactly 4 questions correctly using the same approach:

Probability of answering 4 questions correctly = (number of ways to choose 4 questions correctly) * (probability of choosing the correct answer for each of those 4 questions)^4 * (probability of choosing the incorrect answer for the remaining 2 questions)^(6-4)

(6 choose 4) = 6! / (4! * (6-4)!) = 15

Probability of answering 4 questions correctly = 15 * (1/4)^4 * (3/4)^2

Finally, to find the probability of answering 3 or 4 questions correctly, we sum up the probabilities of both scenarios:

Probability of answering 3 OR 4 questions correctly = Probability of answering 3 questions correctly + Probability of answering 4 questions correctly

This gives us the desired probability.