Double Blind

The choice of lotteries out there always involve negative statistical expectation, but people pay for skew and not expected value. People do this all the time when they purchase insurance, for example.

Very often people in an office will pool capital and spend the proceeds on a lottery.

The question is: Which lottery? I will isolate the definition to the lottery that gives the highest probability of delivering a main prize worth $1,000,000 or over.

The BC49 involves paying $1 for a 1:13,983,816 chance of winning a share of a prize that is guaranteed to be above $2 million.

The Lotto 6/49 involves paying $2 for a 1:13,983,816 chance of winning a share of a prize that is guaranteed to be above $3.5 million.

The Lotto MAX involves paying $5 for a 1:28,633,528 chance of winning a share of a prize that is roughly $10 million at a minimum.

Note the terminology: “a share of a prize” – in that if there are multiple winning tickets, your share of the main prize will be significantly lessened. Thus, it is critical that you choose numbers that others do not. The rest of this analysis assumes this is the case except when explicitly noted below.

The BC49 and Lotto 6/49 are very similar games. As such, if the Lotto 6/49′s main prize is above $4 million then it is always better to choose the Lotto 6/49. One mitigating circumstance is that the Lotto 6/49 is much more likely to share a main prize because it is a national lottery, compared to the relatively unplayed BC49 lottery – if the main prize is slightly over $4 million for the Lotto 6/49 you will likely have better statistical expectation with the BC49.

The other factor is that you can buy two tickets of BC49 compared to one ticket of Lotto 6/49. So to maximize your chances of winning $1,000,000; you would be best to choose BC49.

For an office pool, presumably the allure of a more relevant main prize would be with the Lotto 6/49. The trick is waiting for a sufficiently large prize pool to make it worthwhile – I would suggest waiting until a prize pool of least $28 million is available to maximize expectation.

Lotto MAX is a little more difficult to analyze, but the feature of this lottery is that if the prize pool goes above $50 million, there are million dollar prizes available which you also have a 1:28,633,528 chance winning in each instance. So for example, on February 26, 2010 and November 6, 2009 there was a $50 million main prize up for grabs, but also 10 draws for $1 million each.

A five dollar play in this instance would have a 1:2,603,048 chance of winning $1 million or over, which is roughly the expected value available for a Lotto 6/49 player paying $2 for the chances of winning a $29 million prize, but with the skew being such that your net probability of winning a 7-digit prize is about 5.4 times greater for 2.5 times the cost.

Without this $1 million bonus draw feature, the game itself is only worth playing over the 6/49 if its main prize pool is 5.12 times larger than the Lotto 6/49. (i.e. if the Lotto 6/49 estimated prize is $6 million, the Lotto MAX pool has to be $31 million or over to consider playing, even assuming you are not considering the BC49 for the higher chance of winning a million or over).

So my advice for lottery pool players:

1. It is absolutely essential you choose numbers that others do not.

2. Patience is your friend; do not play every draw and wait for the prize pools to grow sufficiently large, where “sufficiently large” we will define as being greater than $28 million for the 6/49, and $50 million for Lotto MAX.

3. If you MUST play every week, play the BC49 as long as the 6/49 prize pool is under (roughly) $6 million, which assumes the 6/49 jackpot will get split 50% of the time.

4. If the Lotto MAX jackpot is $50 million, consider playing that instead of the 6/49, especially if the 6/49 prize pool is under $10 million. If the 6/49 jackpot is over $10 million, then play that instead UNLESS if you are trying to have a better chance of winning a $1M prize, which in that case choose the Lotto MAX instead.

Lotteries are horrible games of chance simply because there are so many other alternatives that give you much higher chances of winning your desired pot of money. A really simple example would be a single-zero roulette wheel (1-36 and 0); if you put money on a single number and the desired number comes up, you will be paid out 35:1 (i.e. a $1 winning bet would be transformed into a $35 profit plus the $1 bet); if you took $5 and parlayed it four times successfully (and found a casino willing to take the risk in question), you would be $8.4 million dollars richer, but your risk taken would be 1:1,874,161, considerably much better than any of the options above. If your goal was to make exactly $1,000,000, then you would take your $5 and parlay it three times, earning $233,280, at an odds of 1:50,653. You would then structure your $233,280 into 8 equal bets of $29,160 and at this point have at least a 19.7% chance of winning $1,000,000 (for a combined probability of 1:257,340 of winning $1 million playing single-zero roulette from a $5 initial bet).

As you can see, lotteries are horrible at returning money to their customers compared to other games of chance.

This entry was posted on Sunday, April 25th, 2010 and is filed under Commentary. You can follow any responses to this entry through the RSS 2.0 feed. You can skip to the end and leave a response. Pinging is currently not allowed.