A multiple choice test has 10 questions, each of which has 4 possible answers, only one of which is correct. If Judy, who forgot to study for the test, guesses on all questions, what is the probability that she will answer exactly 3 questions correctly? Round your answer to four decimal places.

Solution:

Given, a multiple choice test has 10 questions.

There are 4 possible answers out of which only one is correct.

Judy forgot to study and answered all questions on guesses.

We have to find the probability that she will answer exactly 3 questions correctly.

Total number of options = 10(4) = 40

Total number of correct answers = 10

Probability of picking a correct answer = 10/40 = 1/4 = 0.25

To number of wrong answers = 30

Probability of picking a wrong answer = 30/40 = 3/4 = 0.75

Now, she has to answer 3 questions correctly which means 7 answers are wrong.

By using binomial distribution,

\(\sum_{k=0}^{n}\, ^{n}C_{k}(p)^{k}(q)^{(n-k)}\)

Here, n = 10, k = 3, p = 0.25 and q = 0.75

So, \(^{10}C_{3}(0.25)^{3}(0.75)^{(10-3)}\\=(\frac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{(3\times 2\times 1)(7\times 6\times 5\times 4\times 3\times 2\times 1)})(0.015625)(0.75)^{7}\\=120(0.015625)(0.1335)\)

= 0.2503

Therefore, the probability that she will answer exactly 3 questions correctly is 0.2503

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A multiple choice test has 10 questions, each of which has 4 possible answers, only one of which is correct. If Judy, who forgot to study for the test, guesses on all questions, what is the probability that she will answer exactly 3 questions correctly? Round your answer to four decimal places.

Summary:

A multiple choice test has 10 questions, each of which has 4 possible answers, only one of which is correct. If Judy, who forgot to study for the test, guesses on all questions, the probability that she will answer exactly 3 questions correctly is 0.2503