A jackpot of $29 million! Wouldn't that be nice? That's the anticipated first prize in the upcoming draw on Wednesday June 7, 2006 in Canada's national lottery, Lotto 6/49. Should you, as a rational investor, buy a ticket?

One of the interesting features of lotteries such as Lotto 6/49 is that the expected value of a ticket can and often does rise as the jackpot goes up. The expected value of a game of chance is obtained by multiplying the odds of winning by the amount of the prize. In this case, the odds of winning the Lotto 6/49 jackpot is estimated by the Interprovincial Lottery Corporation to be about 1 in 13,983,816, call it 14 million, multiplied by the value of the jackpot. Of course one also has to account for the possibility of multiple winners, which reduces the size of the jackpot. If I win and someone else picks the same numbers, we split the jackpot equally. Basically everyone who wins is cut in for a share of the jackpot.

A little more background helps complete the analysis. Draws are held every Wednesday and Saturday. If the jackpot is won, it is reset for the next draw at a guaranteed $4 million regardless of the level of ticket sales. Forty-seven percent of the ticket sales go into the so-called Pool Fund and from the Pool Fund comes all the $5 and $10 prizes for 2/6+ and 3/6 correct number combinations (what happens to the other 53% is another story ...). What's left over from the 47% is then divvied into the 4/6, 5/6, 5+/6 and 6/6 number pots. The six number pot - where you get them all correct - is 80.5% of what's left over after taking care of the $5 and $10 prizes.

There's one more kicker. If there is no winning ticket for the jackpot, the jackpot gets rolled into the next jackpot (subject to some mind-bending legal mumbo-jumbo that seems to spread the wealth downwards into the 4, 5 and 5+ pools if the 6/6 pool gets over $25 million).

Now here's the interesting part. Start with an assumption that there are 5 million tickets sold at $2 each for an average draw with a minimum jackpot. About half the ticket sales go into the prize pool, and from that they pay the lower value prizes. You have a 1 in 14 million chance of winning $4 million, subject to sharing it with one of the other 3,999,999 tickets sold. Your expected value (looking only at the jackpot) is 1/14,000,000 times $4 million, times 4,000,000/14,000,000, or about the square root of nothing (actually, it's about 8 cents on a $2 ticket). Not the best payout.

Nobody wins for a few weeks and the jackpot grows to $29 million. Now, assume that the same number of tickets are sold (I know, this probably won't be the case, but humour me) - 4 million. Now you have a 1 in 14 million chance of winning $29 million, subject to sharing it with one (or more) of the other 3,999,999 tickets sold.

Your expected value (looking only at the jackpot) is now 1/14,000,000 times $29 million, times 4,000,000/14,000,000, or about $0.60 on your $2 ticket. It just grew by 7.25 times when compared to the $4 million jackpot!

More realistically, there will be more tickets sold as the jackpot grows. Say there's twice as many tickets sold. Your odds of selecting the correct six numbers never changes - that is independent of the size of the jackpot and also independent of the number of other winners. So you still have a 1 in 14 million chance of winning. But now the odds of sharing it with someone else (or a bunch of someone elses) has doubled, so instead of having an expected value of 7.25 the base amount, it's half that or 3.625 times better than the base case-maybe around 30 cents on your two dollar ticket.

The reason that you have a higher expected value as the jackpot rolls over and increases is that the poor suckers who played last week, and the week before, and maybe even the week before, now have no chance to win with their expired tickets, but they have effectively contributed some of their ticket price to your prize pool through the rollover. New money coming in gives you the same expected value as before but the rollovers are what you're after if you are seeking to maximize your expected value.

So is it a good investment? No, but at least it's better than what it was before!mortgage calculator