How many people are needed in a team to ensure one is at work?

Question

How does one estimate the probability that at least one person from a team is in the office during working hours (9 am - 5 pm)? My assumptions are:

• There are 3 people in the team
• There are 253 working days in the year
• Each person has 38 working days out of the office (annual leave + meetings)
• Days out of the office are randomly taken
• When not out of the office the person is available (ignore sick leave etc.)

The question is a real life question that is needed to work out the amount of time that the office is not staffed. The extension to the question above is: How many people are needed to increase the probability to say 95%?

I have looked at the coupon-collector's problem, but I have little experience in calculating probabilities and so I am unsure how to apply it to this problem. What are the steps required?

Answer 1

Hints:

• Calculate the probability that a fixed person is working on a fixed day.
• Calculate the probability that none of the $3$ persons are working on a fixed day.
• Calculate the probability that at least one of the $3$ persons is working on a fixed day.
• Do the same for $n$ instead of $3$ to get some view on how this depends on the size of the team.
• Let $X_i=1$ if at least one person is working on day $i$ and $X_i=0$ otherwise. Then $X_1+\cdots+X_{253}$ equals the number of days that at least one is at work. Find its expectation and draw conclusions about the fraction you mention.