Question

A multiple choice test has 7 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?

Answer 1

Each question is a random variable, with probability 1/4 of success (only one answer over four is correct). We repeat this experiment 7 times, hoping to succeed three times.

This is exactly the case where Bernoulli distribution is useless. We use it when we repeat identical experiments $n$ times, expecting $k$ successes, with a success probability of [/tex] p [/tex] (which implies a failure probability of $1-p$. That probability is

$P (\text{k successes over n trials}) = \binom{n}{k} p^k (1-p)^ {n-k}$

where the binomial coefficient is defined as

$\binom{n}{k} = \cfrac{n!}{k!(n+k)!},\quad \text{where } n! = n(n-1)(n-2)\ldots 3\cdot 2$

If we plug our values, we have

$P (\text{3 successes over 7 trials}) = \binom{7}{3} 0.25^3 (0.75)^{4} \approx 0.17$