Question

Question 61 pts Richard has been given a 12-question multiple-choice quiz in his history class. Each question has three answers, of which only one is correct. Since Richard has not attended the class recently, he doesn't know any of the answers. Assuming that Richard guesses on all 12 questions, find the probability that he will answer at least 2 questions correctly. Round your answer to the nearest thousandth.

Answer 1

Answer:

0.946 = 94.6% probability that he will answer at least 2 questions correctly.

Step-by-step explanation:

For each question, there are only two possible outcomes. Either he answers it correctly, or he does not. The probability of answering a question correctly is independent of other questions. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

12 questions.

This means that $n = 12$

Each question has three answers, of which only one is correct.

This means that $p = \frac{1}{3} = 0.3333$

Assuming that Richard guesses on all 12 questions, find the probability that he will answer at least 2 questions correctly.

Either he answers less than 2 questions correctly, or he answers at least 2 questions correctly. The sum of the probabilities of these outcomes is decimal 1. So

$P(X < 2) + P(X \geq 2) = 1$

We want $P(X \geq 2)$. So

$P(X \geq 2) = 1 - P(X < 2)$

In which

$P(X < 2) = P(X = 0) + P(X = 1)$

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = x) = C_{12,0}.(0.3333)^{0}.(0.6667)^{12} = 0.008$

$P(X = x) = C_{12,1}.(0.3333)^{1}.(0.6667)^{11} = 0.046$

$P(X < 2) = P(X = 0) + P(X = 1) = 0.008 + 0.046 = 0.054$

$P(X \geq 2) = 1 - P(X < 2) = 1 - 0.054 = 0.946$

0.946 = 94.6% probability that he will answer at least 2 questions correctly.