Final answer:
The probability of 25 members checking in during the first hour can be determined using the Poisson distribution given the average rate of check-ins, but the provided information is inconsistent with the scenario in question.
Step-by-step explanation:
The probability of 25 people checking in during the first hour when the average is 20 can be computed using the Poisson distribution, which is used to model the number of events that occur within a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.
To find the probability, we use the Poisson probability mass function (PMF):
P(X=k) = (λ^k * e^-λ) / k!
Where λ is the average number of events (in this case, 20 members), k is the number of events we are interested in finding the probability for (in this case, 25 check-ins), and e is the base of the natural logarithm. However, the student provided information about a gym check-in scenario does not align with the average provided concerning phone calls in the reference information, which would also require knowledge of the time period specifically defined for 'the first hour' mentioned for the gym to accurately calculate this probability.