Answer to Question #344871 in Statistics and Probability for Akshay

Question

On a five questions, multiple choice test, there are 5 possible answers of which 1 is correct. If a student guesses randomly and independently, what is the probability that she is correct only on 2 questions?

Answer 1

The probability that the student guesses the right answer $p = \frac{1}{5}=0.2,$ that she doesn't guess $q = \frac{4}{5}=0.8.$

We have a Bernoulli trial - exactly two possible outcomes, "success" (the student guesses the right answer) and "failure" (she doesn't guess) and the probability of success is the same every time the experiment is conducted (the student answers a question).

The probability that she answers correctly k times

$P(X=k)=\begin{pmatrix}n\\k\end{pmatrix}\cdot p^k\cdot q^{n-k}=\\ =\begin{pmatrix}5\\k\end{pmatrix}\cdot 0.2^k\cdot 0.8^{5-k}=\\ =\cfrac{5!}{k!\cdot(5-k)!}\cdot 0.2^k\cdot 0.8^{5-k}.$

The sought probability that she is correct only on 2 questions:

$P(X=2)=\begin{pmatrix}5\\2\end{pmatrix}\cdot 0.2^2\cdot 0.8^{5-2}=\\ =\cfrac{5!}{2!\cdot3!}\cdot 0.2^2\cdot 0.8^{3}=0.2048.$