The biology test has ten questions. Each question has four possible answers, but only one of them is correct. Jenda didn't study and now he randomly selects answers. What is the probability that he won't fail the test, given that he needs to answer at least $3$ questions correctly to pass?
Instead of studying biology, Jenda calculated the relevant probability:
(1) First, he calculated the probability of getting all answers wrong. "This shouldn't happen to me!" He calculated the probability using the binomial distribution: $$P_0={10\choose10}\cdot0.75^{10}\cdot0.25^0=0.0563$$
(2) Next, he calculated the probability of getting $9$ answers wrong and $1$ correct: $$P_1={10\choose9}\cdot0.75^9\cdot0.25^1=0.1877$$
(3) He further calculated the probability of getting $8$ answers wrong and $2$ correct: $$P_2={10\choose8}\cdot0.75^8\cdot0.25^2=0.2816$$
(4) He also calculated the probability of getting $7$ answers wrong and $3$ correct: $$P_3={10\choose7}\cdot0.75^7\cdot0.25^3=0.2503$$
(5) The probability that Jenda won't pass the test is: $$P=P_0+P_1+P_2+P_3=0.7759$$ (6) So, the probability that Jenda will pass the test, i.e., answer at least $3$ questions correctly, is: $$1-0.7759=0.2241$$
He found this probability too low, so he decided to go back to studying biology.
Jenda failed the biology test. Did he at least manage to correctly solve this probability problem?
Jenda solved the problem correctly; he might be more of a mathematician than a biologist.
Jenda made a mistake in step (1). It holds $0.25^0=0$. Therefore $P_0=0$.
Jenda made a mistake in step (5). The probability that he won't pass the test is the sum of $P_0+P_1+P_2=0.5256$. The probability that he will pass is $1-0.5256=0.4744$.
Jenda made a mistake in step (6). The probability that he will pass the test is the one already calculated in step (5), which is $0.7759$.