From Probability and Statistics for Engineers and Scientists, 9th edition:
According to a study published by a group of University of Massachusetts sociologists, approximately $60\%$ of the Valium users in the state of Massachusetts first took Valium for psychological problems. Find the probability that among the next $8$ users from this state who are interviewed,
(a) exactly $3$ began taking Valium for psychological problems;
(b) at least $5$ began taking Valium for problems that were not psychological.
I got part (a) right, but I'm unsure why my approach for part (b) isn't working. My approach:
Let $X$ denote the number of users who began taking Valium for non-psychological reasons. Then the problem is asking us to find $P(X\geq5)$. Note that the probability that a Valium user first took Valium for non-psychological problems is $40\%$ (which is $100-60\%$).
In terms of binomial distributions, this would be:
$${8 \choose 5}(0.4^5)(0.6^3)+{8 \choose 6}(0.4^6)(0.6^2)+{8 \choose 7}(0.4^7)(0.6)+{8 \choose 8}(0.4^8)(0.6^0)$$
Calculating this, I got an answer of $0.1736704$.
The book did the following:
Now, someone correct me if I'm wrong, because I feel like I'm going crazy here—didn't they calculate the probability of at least $5$ users taking Valium for problems that were psychological? I just calculated that and got their answer. But the problem asks for not psychological.
Your working is right.
They computed the probability that at least $5$ began taking Valium for psychological problem.