Mark's Blog

At work, we have a weekly 50/50 draw, the proceeds of which go to an annual Christmas party. I didn't go last year but I'll usually buy a few tickets for this sort of thing. I have been playing for a while (and won $57 last week), but I decided to analyze the draw a little more closely to see what the actual odds are. There are three ways to purchase tickets, 1 for $1 ($1/ticket), 3 for $2 ($0.67/ticket), and 25 for $5 ($0.20/ticket). As the name of the draw suggests, half of the money collected is given as the prize for winning the draw.

First of all, I asked the people selling the tickets what the typical distribution of ticket buyers was, in order to help me estimate the odds in winning. It turned out that most ticket buyers buy 3 and that there are typically about 5-10 people who buy 25. A similar number, 5-10 buy 1 ticket. There are usually about 60-80 people buying tickets. Then, I made a spreadsheet to calculate the expected values. As an example, if 10 people buy 1 ticket, 60 people buy 3 tickets and 10 people buy 25 tickets, then the expected return, is about -80% for buying 1 ticket, -69% for 3 tickets, and +2% for buying 25 tickets. It's somewhat amazing that in a lottery giving back only 50% of the prize money, that there can actually be a positive expectation. Of course, this is due to the deep discount given on buying 25 tickets at a time.

With most gambling opportunities, any time you can make a bet that is favorable it is in your interest to do so as much as possible. Not true for this lottery though -- the odds get better for the 25-ticket buyers the fewer of them there are, and the odds get worse for the 1- or 3-ticket buyers the more of them there are. This leads to the somewhat surprising conclusion that for those of us buying 25 tickets, buying an extra 25 tickets actually decreases our expected value. Possibly, even from a positive expectation to a negative expectation.

So how is this possible, you ask? The key ratio to look at is the prize money divided by the number of tickets sold. If the cost of your ticket is less than this figure, then your expectation is positive, and if it's more, your expectation is negative. Each person who buys a ticket makes this ratio a little bit closer to half of the cost of their ticket, due to the fact that it's a 50/50 draw. And, for most practical numbers of ticket-buyers, it turns out that the ratio is usually between 0.10 and 0.33 (i.e. half of the average cost of 25 tickets and of 3 tickets), therefore any marginal purchase of 1 or 3 tickets helps everybody, and the marginal purchase of 25 tickets hurts everybody.

These calculations do not apply to lotteries such as 6/49 or Super 7 since all tickets purchased cost the same amount. In this case, the expectation cannot change as it does in the 50/50 draw as discussed. And of course, the expectation is always negative. Buying more tickets may increase the absolute amount lost or won, but it does not change the expectation per dollar wagered.