Question:
Amy is a private coach in a gym. In order to encourage her clients to come to her training sessions regularly and on time, she decides to establish a fund of $1000$ dollars from which she will pay an amount, $q$, at the end of the year to each of her $20$ clients who achieve perfect attendance over that year.
Suppose each client has a $1 \%$ chance of achieving perfect attendance during any given year, independent of any other client. Determine the maximum value of $q$ such that there is a less than $1 \%$ chance that the $ \$1,000$ fund will be inadequate to cover all payments for perfect attendance.
Proposed Solution:
As $q$ is based on less than $1 \%$ chance of $\$1,000$ being inadequate, that should translate to a probability of less than $1 \%$ of perfect attendance.
Let X be a binomial random variable wit $n = 20$, $p = 0.01$ representing the number of perfect attendances.
P(X=1) = 16.5%, P(X=2)=1.58% P(X=3)=0.09%
Since, $P(X=3)<1$ $\%$, $q$ should be based on $3$ perfect attendances. Therefore $q$ should be $1000/3 = 333.33$.
Does anyone agree with this solution?
You should actually compute the value for which $P(X > k) \leq 1\%$
$P(X>1) = 1.69\%\;\; P(X>2) = 0.10\%,$
proceed.....