Binomial distribution calculator

You are sitting a test with five questions, each with four options, and you haven’t revised at all, so you are guessing. What is the probability that you get exactly two questions right? This is what the binomial distribution is for: out of n independent attempts, each of which succeeds with probability p, it gives the chance of exactly k successes.

Enter the probability p as a percentage. The calculator also accepts fractions, so a quarter can be entered as 25 or 100/4, and a third as 100/3.

How it works

The calculator uses the formula

$$P(X=k)=(\genfrac{}{}{0ex}{}{n}{k})\cdot p^k\cdot (1-p{)}^{n-k}$$

The probability of getting two specific questions right and the other three wrong is ${\left(\frac{1}{4}\right)}^2\cdot {\left(\frac{3}{4}\right)}^3=\frac{27}{1024}$. However, the two correct answers can be placed among the five questions in $(\genfrac{}{}{0ex}{}{5}{2})=10$ ways, so we multiply this probability by ten:

$$P(X=2)=(\genfrac{}{}{0ex}{}{5}{2})\cdot {\left(\frac{1}{4}\right)}^2\cdot {\left(\frac{3}{4}\right)}^3=10\cdot \frac{27}{1024}=\frac{270}{1024}\approx 0.2637$$

Guessing, you get exactly two of the five questions right with a probability of about 26.37%. If you are interested in “at least once”, the quicker option is the at least once probability calculator. Sums such as “at least three” are found by adding the results for three, four and five successes, as the article on the binomial distribution shows.