I want to calculate the probability to win in a certain lottery. These are the rules:
In the lottery, participants can participate individually or as a group. Groups can have up to 200 members.
Each lottery ticket has a different number. Numbers are three digits long. Each digit can be any of 0 to 9. Thus, numbers run from 000 to 999 and there are one thousand lottery tickets.
When an individual participates, he wins a price when his number is drawn. When a group participate, they all get different numbers, and the group wins a price when one of their numbers is drawn.
(The value of the price is irrelevant for the question. For the question, winning as an individual is as good as winning as a member of a group. Imagine the price being a journey for all winners.)
There can be between 1 and 20 prices, all of equal value. That is, between one and twenty numbers are drawn. Numbers differ between drawings.
When the numbers are drawn, each digit is drawn individually. First, the first digit of the number is drawn from an urn with balls numbered 0 to 9. After the first digit has been drawn, the drawn ball is returned to the urn and the second digit is drawn. Again the ball is put back and the third digit is drawn. If there is more than one price, the third ball is returned to the urn and the next numbers are drawn in the same way as the first.
Therefore it is possible that the same number is drawn multiple times. If this happens, the drawing is counted as a blank and another number is drawn. That is, each number can only win once, even if it is drawn multiple times.
The main question I'm trying to answer is whether the probability for a group win differs between the following two cases:
A: The lottery ticket numbers of all members in a group begin with the same digit, e.g.
853 812 877
B: The lottery ticket numbers of all members in a group begin with different digits, e.g.
755 429 186
For the question I assume a group of three persons whose numbers differ in the second and third digit. That is, only the first digit is relevant for the question.
I'm interested in the answer, of course, but I'd like to understand how to calculate it. I have some basic knowledge in calculating probablities for combinations and permutations, with and without returning the ball to the urn, but I find myself unable to wrap my head around the group win problem.