College Algebra, Chapter 10, 10.4, Section 10.4, Problem 14

Five independent trials of a binomial experiment with probability of success $p=0.7$ and probability of failure $q = 0.3$ are performed. Find the probability of each event.

At most three failures

Recall that the formula for the binomial probability is given by

$C(n,r) p^r q^{n-r}$

If the event should have at most three failures, then it means that the event must have at least two successes. Thus, we have


$\begin{equation} \begin{aligned} P(\text{At least two successes}) =& P (\text{exactly 2 successes}) + P (\text{exactly 3 successes}) + P(\text{exactly 4 successes}) + P(\text{exactly 5 successes}) \\ \\ =& C(5,2) (0.7)^2 (0.3)^{5-2} + C(5,3) (0.7)^3 (0.3)^{5-3} + C(5,4) (0.7)^4 (0.3)^{5-4} + C(5,5)(0.7)^5 (0.3)^{5-5} \\ \\ =& 0.1323+ 0.3087 + 0.36015 + 0.16807 \\ \\ =& 0.96922 \end{aligned} \end{equation}$