Exactly One Success

Idea

“Exactly one success” means the outcome has one favorable case and all the others are not successful. In statistics, this is usually handled by a binomial model when trials are independent and each trial has the same success probability.

Method

1. Identify the number of trials: call it (n).
2. Identify the success probability: call it (p). Then failure probability is (1-p).
3. Count the ways to place the single success: there are (n) possible positions, so use (\binom{n}{1}=n).
4. Write the probability: [ P(\text{exactly one success})=\binom{n}{1}p(1-p)^{n-1}. ] Since (\binom{n}{1}=n), this is also [ np(1-p)^{n-1}. ]
5. Simplify the expression carefully, especially powers and fractions.

Check

Make sure the answer is a probability, so it should be between 0 and 1. If the result is greater than 1 or negative, recheck the exponent, the binomial coefficient, and whether you used (1-p) for failures.