# Chance of randomly guessing 21 questions right out of 50 with 4 multiple choice.

Lets say a person decided to randomly fill in a scantron of 50 questions with 4 choices each. After submitting it to be graded, the result was 42% correct. How would we figure out the probability of this occurring completely by chance?

This problem actually came up while I was watching a show. I would assume that a probability test is to be used here. Observed is 21, while the expected is 12.5 questions. How would one go from here?

A correct answer has a probability of $1/4$ for an individual question, and $3/4$ for an incorrect answer.
There are ${50 \choose 21}$ ways to select the 21 correct answers out of 50 questions. So the final probability should be $${50 \choose 21} (1/4)^{21}(3/4)^{29}$$ This works out to about 0.3%
• @SeitokaiShino: as an approximation, adding another correct reduces the probability by about a factor $3$ as the combinatoric factor doesn't change much. The sum of $1+\frac 13+\frac 1{3^2}+\dots=\frac 32$, so the total will be about $\frac 32$ times the value above. – Ross Millikan Mar 5 '14 at 18:56