From SOA Sample #199:
A company has five employees on its health insurance plan. Each year, each employee independently has an $.80%$ probability of no hospital admissions. If an employee requires one or more hospital admissions, the number of admissions is modeled by a geometric distribution with a mean of $1.50$. The numbers of hospital admissions of different employees are mutually independent. Each hospital admission costs $20,000$. Calculate the probability that the company’s total hospital costs in a year are less than $50,000$.
I know how to find part of the solution, which involves finding the binomial for $0, 1$, and $2$ employees being hospitalized. I also know that the probability of the geometric distribution here is $2/3$. What I don't understand is what the geometric distribution means in this case. Until now, I understood that geometric distribution is either the number of trials to get to the first success, or the number of failures before the first success, but neither of those seem to be applicable here.
Is this some different definition of the geometric distribution, and if so what is it, or does the standard definition apply, and if so, how?