Hi, I was wondering if you could help me out with a problem I have:

I need to work out the probability that someone will be able to complete a minimum of 10 hours of rehearsals in one week. The probability that this person is free each day is 60% on a weekday and 30% on the weekend. There are 3 hours of rehearsal time available every day except Tuesday and Wednesday when there are 4.

If available on a given day then they can use all the time available to them.

I think it’ll be easier to turn the problem around by asking what the probability is that a person will not make 10 hours of rehearsal in a week, and to subtract this probability from 1.

There is no combination of weekdays that will give at least 10 hours in total if the person attends 0, 1 or 2 days in a week. If they attend for more than 3 days in a week, the total will always exceed 10 hours. For exactly 3 days, the total will be less than 10 hours only if both Tuesdays and Wednesdays are excluded.

Each probability of attending exactly 0, 1, 2 and 3 (with no Tuesday or Wednesday) can now be evaluated separately from first principles without much difficulty. The sum of these four probabilities will give the overall probability that the person will make less than 10 hours in a week.