Answer to Question #337046 in Statistics and Probability for dabb

Question

A multiple choice quiz has five questions, each with four answers, a through d. If you just guess on all of the questions, what is the probability that you get exactly four right?

Answer 1

We have a Bernoulli trial - exactly two possible outcomes, "success" (I guess on the correct answer) and "failure" (I don't guess on the correct answer) and the probability of success is the same every time the experiment is conducted (I guess on an answer), $p=\cfrac{1}{4}=0.25, q=1-0.25=0.75, n=5.$

The probability that I guess on k correct answers

$P(X=k)=\begin{pmatrix}n\\k\end{pmatrix}\cdot p^k\cdot q^{n-k}=\\ =\begin{pmatrix}5\\k\end{pmatrix}\cdot 0.25^k\cdot 0.75^{5-k}=\\ =\cfrac{5!}{k!\cdot(5-k)!}\cdot 0.25^k\cdot 0.75^{5-k}.$

The probability that I guess on 4 correct answers

$P(X=4)=\cfrac{5!}{4!\cdot1!}\cdot 0.25^{4}\cdot 0.75^{1}=0.0146.$