# Expected value of a lottery ticket

1. Dec 2, 2011

### elementbrdr

I am going to try to keep this short, so please advise whether I need to provide more detail for my question to make sense.

In calculating the expected value of a lottery ticket, one must consider the possibility that more than one ticket is sold bearing the winning combination. One way to account for this is to calculate the individual probabilities of a winning combination being shared by 1, 2, 3, ... , n tickets. E.g. the method used in the following link: http://dematerialism.net/expval.htm.

I was wondering whether it is also possible to account for multiple winners using the following method: V = (1/c * j) / (t/c)
t = tickets sold
j = jackpot value
c = combinations of tickets (not really a variable, but using it as such for readability)

Green reflects odds of winning times jackpot. Red reflects that this expected value should be divided by the mean number of tickets with the same combination of numbers (the mean number of people with whom any winner should expect to share a prize).

I recognize that this would inflate the value of the jackpot for t/c<1. However this could be accounted for by defining a formula that omits division by t/c when t/c<1.

However, the fact that this adjustment is required makes me nervous about the overall reasonableness of this approach.

2. Dec 3, 2011

### Office_Shredder

Staff Emeritus
Do you realize your formula is just j/t? Which is the expected value of the lottery assuming exactly one person wins it.

In general, if X and Y are random variables, E(XY) is NOT equal to E(X)E(Y) so you can't just split up multiplications

3. Dec 3, 2011

### elementbrdr

Thanks for your response. I did not notice that the formula could be simplified before posting. Thanks for the clarification.

After further work on this problem, I have learned that one of my key assumptions -- that combinations on tickets are uniformly distributed -- is false. So the entire formula is clearly incorrect.

At this point I'm just trying to figure out whether the overall approach of dividing by the correct value for the mean number of winning tickets with the same combination of numbers works.

I wasn't able to put your statement regarding multiplication of expected values in context. Sorry. Could you explain more specifically to how that relates to what I'm trying to do?

Thank you.