Probability that a vaccine works on all patients to which it is administered I have the following problem: > The probability of success of a vaccine is 0.8. The vaccine is administered to 10 patients. What is the probability that none of them suffer the sickness. Then the solution was determined to be: $$ P(0) = \binom{10}{10} \cdot 0.8^{10} \cdot 0,2^{10} $$ I understant that $\binom{10}{10}$ is any combination of 10 patients and $0.8^{10}$ is the probability of the vacination working with all patients but why include the the probability of it not working on the patients $(0.2^{10})$?

It should be $0.2^0$, which is to say, it isn't really included in the actual calculation.

The exponent of $0.8$ and the exponent of $0.2$ should add up to the total number of patients, whcih is $10$. The exponent of $0.8$ should represent the number of patients for which the vaccine worked (i.e. the number of events of probability $0.8$ that occurs), and the exponent of $0.2$ should represent the number of patients for which the vaccine didn't work (i.e. the number of events of probability $0.2$ that occurs).