"A lottery consists of 100 tickets, labeled 1,2,3,4,...100, three of which are "winning numbers". You buy 4 tickets. Calculate the probability, p, that you have at least one winning ticket. Try deriving the solution via Poincare's theorem"
Poincare's Theorem is given by: $ Pr(\bigcup\limits_{i=1}^{n} A_{i})= \sum_{i=1}^{n} (-1)^{i+1}S_{i} $
where $S_{j}=\sum_{i_{1}<....<i_{j}}Pr(A_{i1}\cap A_{i2}...\cap A_{ij})$
So $A_{i}$ = {you purchase a winning ticket} ; $i=1,2,3$
How would you apply Poincare's Theorem here? I've never used it before on an actual example. I only know the theory but I would like to know how it works on a example.
Applying normal combinatorics I get approximately $0.1164$ as a solution.
The event "at least one winning ticket" is a union of four events: exactly $i$ winning tickets, where $i$ varies between $1$ and $4$. The exclusion-inclusion principle (which you call "Poincare's theorem") then can be used to calculate the probability of the union as follows: The probability to win exactly one ticket minus the probability to win exactly two tickets plus the probability to win exactly three tickets minus the probability to win exactly four tickets. I'm sure you can do the calculation.