My work is having a raffle and I'd like to know if my expected value goes up if I purchase more than 1 ticket?
Here are the rules:
A ticket costs $5.
Two tickets will be randomly drawn.
The winner of the first drawing gets 50% of the pot and their winning ticket is NOT returned to the pool of tickets.
The winner of the second drawing gets the other 50% of the pot.
In the case that the same person bought both winning tickets, they are allowed to win both prizes and thus get 100% of the pot.
To me it seems like I would be best off if I buy 2 tickets, but I'm not sure how to do the math behind it.
Also, I know the # of total tickets must play a role in this equation, so I'd like to see the answer for the possibility that 48 other people have bought tickets and the possibility that 98 other people have bought tickets.
So for case 1 (48 other people bought tickets):
If I bought 1 ticket, my odds of winning the first drawing is 1/49 and my odds of winning the second is 0/48 if I won the first and 1/48 if I lost the first.
I'm not sure where to go from here or how to transfer this logic to if I bought 2 tickets.
Best Answer
So the pot is $50\times5=\$\,250$. If you buy one ticket then your probability of winning is $2/50$ and expected winning is: $$ \frac2{50}\times 125=\$\, 5 $$ (which is as expected, as your cost is also $\$\,5$ and is is a zero-sum game). If you buy two tickets then the expected win for each of them is also $\$\,5$, for a total of $\$\,10$ (expected values add up).
Therefore, in terms of expected value alone, buying any number of tickets does not make a difference.
In terms of utility, well it depends. You may value the fact of winning, in which case buying more tickets is appealing, or be risk-averse and buy one ticket (or none at all).
Note that the expected value criteria is just an indication, as the game is a one time thing.