So I'm teaching myself some probability and stats which go beyond the discussion of my behavioural research/stats book, and I'm stuck on one problem.
This following problem has been taken from Statistics and Probability for Scientists and Engineers by Anthony Hayter, 2012. Chapter 2, Exercise 2.1.4, Q.no. is 2.1.11, page number 81.
A consultant has six appointment times that are open, three on Monday and three on Tuesday. Suppose that when making an appointment a client randomly chooses one of the remaining open times, with each of those open times equally likely to be chosen. Let the random variable X be the total number of appointments that have already been made over both days at the moment when Monday’s schedule has just been completely filled. (a) What is the state space of the random variable X? (b) Calculate the probability mass function and the cumulative distribution function of X.
I was able to find out (a) which is X = {3,4,5,6} but I am having trouble with (b). The answer for the PMF is 1/20, 3/20, 6/20, 1/2.
What I did: I tried to work it out, and I feel like the denominator might have 6C3 to account for the 20 since the selection is out of the 3 available appointments on Tuesday and total number of appointments which could have been chosen is 6.
I did (3C0)/(6C3) to get 1/20 and (3C1)/(6C3) to get 3/20, but couldn't get 6/20 and 1/2, so I'm sure I'm doing something wrong.
Hint:
In order for all the Monday appointments to be filled on day $X$, there must have been $2$ Monday appointments and $X-3$ Tuesday appointments on the first $X-1$ days, followed by a Monday appointment on day $X$, for $3 \le X \le 6$.