Most of the problems from this section of my textbook are on this site, but this one is missing. I'd like the community to verify the answer for me, please.
According to an advertising study, $15\%$ of television viewers who have seen a certain automobile commercial can correctly identify the actor who does the voice-over. Suppose that ten such people are watching TV and the commercial comes on. What is the probability that at least one of them will be able to name the actor? What is the probability that exactly one will be able to name the actor?
Here is my solution:
$P$(at least one viewer recognizes actor) $=$
$1 - P$(no viewers recognize actor) $=
1-0.85^{10} = .803.$
$P$(only the first viewer recognizes actor) $ = .15^1\cdot 0.85^9 = .0347.$
Multiply result by $10$ because there are $10$ possible ways to choose one viewer from a group of $10$ viewers, so
$P$(exactly one viewer recognizes actor) $ = .347$
Bonus
$P$(only the first two viewers recognize actor) $=
.15^2\cdot 0.85^8 = .0061 .$
Multiply result by $45$ because that's the number of combinations of $2$ viewers from a group of $10$ viewers, so
$P$(exactly two viewers recognize actor) $ = .276$.