Suppose a student takes a $10$-question true/false exam and guesses at every question.
What is the probability that…
a) the student answers every question incorrectly?
b) the students answers at least one question correctly?
c) the student answers exactly 5 questions correctly?
So far I have solved a. I got $(1/2)^{10}$.
For b I'm not sure what to do.
For c would the answer be $(1/2)^5$?
Best Answer
a) Is fine.
b) Notice that the complementary event of $A =$"at least one right" is $\bar A =$"all wrong". Hence $$P(A) = 1-P(\bar A) = 1-\left(\frac{1}{2}\right)^{10}.$$
c) No. You have to count all the spots where you got them right. Notice that there are $\binom{10}{5}$ ways to choose the questions that are correct. We also know that $5$ are wrong (with chance $1/2$ each time) and $5$ are right (with chance $1/2$ each time). Hence $$P(\text{Exactly 5 right}) = \binom{10}{5}\left(\frac12\right)^5\left(\frac12\right)^5.$$
The number of correct answers in $10$ tries follows a binomial distribution.