Do Not Play the Lottery Unless You Are a Millionaire

Yesterday the Lotto 6/49 jackpot was estimated at 49 million dollars. This jackpot was so unusually high, that its size was mentioned on the evening news last night. A quick calculation showed that the expected value of one lottery ticket was more than $3.50, assuming that I can avoid picking the same numbers as someone else. Given that a ticket only costs $2, it seemed like the smart move was to go out and buy a lottery ticket.

I bought a lottery ticked yesterday. I did not win the lottery. In fact, I got zero numbers correct.

Still, I thought it was a good investment. After all, the expected value was positive. However, after some discussion with a few skeptical friends, I decided to check with our pal, the Kelly criterion.

Knowing when to play the lottery is relatively clear. You play when the expected value of a ticket is more than the cost. The Kelly criterion tells you how much to play by calculating what fraction of your bankroll you should bet when the odds are in your favour. Generally, betting more than twice the Kelly fraction will eventually lead to gambler’s ruin, even if the odds are in your favour.

So last night, after the draw, I did the calculation of what fraction of my bankroll I should bet. This number is small. This number is very small. This number is so small, a better question to ask is, “how large should my bankroll be before one $2 lottery ticket is the optimal investment to make?” Assuming that the big jackpot is the sole payout, and assuming you will not share that jackpot, then for one $2 Lotto 6/49 ticket to be the optimal investment, your bankroll ought to be about 65 million dollars.

I feared the Kelly fraction would be small, but this was smaller than I expected. I found an article on the internet that came to similar conclusion about the Powerball lottery in the United States. In that case you ought to be a billionaire before you buy a Powerball ticket.

There is a nice chart at the end of that article that plots the bankroll needed to play at a given jackpot level. Below I generated similar plots for the bankroll needed to buy one $2 Lotto 6/49 ticket at various jackpot levels.

Below is the bankroll needed to buy one $1 Lottario ticket at various jackpot levels.

Notice that if the Lotto 6/49 jackpot is below about $28 million, you should never play, and if the Lottario jackpot is below about $4 million you should not play it. In these cases, the expect value of a lottery ticket is less than the price of the ticket.

So far, I have been ignoring the other prizes that are available for matching some of the numbers. These prizes will increase the Kelly fraction and decrease the bankroll needed before you should play. Unfortunately, it is hard to estimate the value of these prizes beforehand because most of them are based on the number of tickets sold. Using the published prize numbers from yesterday’s drawing, I calculated that you only need a bankroll of about 29 million dollars to play the Lotto 6/49 at yesterday’s jackpot values.

Although I have made several simplifying assumptions, the general conclusion appears to be that you should not play the lottery until you are a millionaire.