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 [tex] n [/tex] times, expecting [tex] k [/tex] successes, with a success probability of [/tex] p [/tex] (which implies a failure probability of [tex] 1-p [/tex]. That probability is
[tex] P (\text{k successes over n trials}) = \binom{n}{k} p^k (1-p)^ {n-k} [/tex]
where the binomial coefficient is defined as
[tex] \binom{n}{k} = \cfrac{n!}{k!(n+k)!},\quad \text{where } n! = n(n-1)(n-2)\ldots 3\cdot 2 [/tex]
If we plug our values, we have
[tex] P (\text{3 successes over 7 trials}) = \binom{7}{3} 0.25^3 (0.75)^{4} \approx 0.17 [/tex]
