Millions of Americans play the lotto each year as an innocent way to gamble their savings on winning some money and if they lose, help to fund government programs like schools. Some studies coupled with an excellent report by John Oliver show that States actually anticipate lottery earnings and instead of adding that extra money on top of the base budget for programs like schools, they just replace the base budget with lotto money and route the rest of the budget towards other Stately concerns. But this post is not about public policy but about gambling because, after all, the lotto is definitely a game of chance.

I have a few friends who play Lotto Texas somewhat religiously and I’ve always been skeptical of the game. So, I decided to actually calculate how much money they should expect to win per ticket they buy. The first step, collect data.

Winnings Structure (as of today)

The rules of Lotto Texas are quite simple:

  • choose 6 numbers
  • each number can be a number from 1 to 54 (inclusive)
  • each number you choose must be unique (i.e. don’t choose a number you’ve already chosen)
  • order doesn’t matter

If you play the base game, you spend \$1 per ticket. If you play the Extra! game, you pay \$2 per ticket.

The potential earnings are below:

Number Correct Prize Amount Total Prize w/Extra!
6 of 6 $12 million $12 million
5 of 6 $2,084 $12,084
4 of 6 $54 $154
3 of 6 $3 $13
2 of 6 N/A $2

Expected Value

The expected value of something is an idea in mathematics that tells you a value you should expect given enough tries. In our lotto case, we’re trying to find the expected value of a ticket (i.e. if we played the lotto enough times and calculated our earnings and losses, what would be the average value of each ticket we invested in?). For example, if you buy a \$2 ticket, but the expected value of that ticket is only \$1, then over time, you should expect to loose a dollar every time you play - that’s a sucker’s bet. So what’s the expected value of a Lotta Texas ticket? Let’s find out.

Since there’s 54 numbers to choose from and only 6 to choose, we have 25,827,165 ways to choose 6 numbers. Winning the \$12 million means we get all 6 of those numbers right which gives us the probability $\frac{1}{25, 827, 165}$. Below are the rest of the probabilities:

$\mathrm{P}(\mathrm{5\ of\ 6}) = \frac{\binom{6}{5} \binom{48}{1}}{\binom{54}{6}} = \frac{288}{25,827,165} = \frac{32}{2,829,685}$
$\mathrm{P}(\mathrm{4\ of\ 6}) = \frac{\binom{6}{4} \binom{48}{2}}{\binom{54}{6}} = \frac{16,920}{25,827,165} = \frac{376}{573,937}$
$\mathrm{P}(\mathrm{3\ of\ 6}) = \frac{\binom{6}{3} \binom{48}{3}}{\binom{54}{6}} = \frac{345,920}{25,827,165} = \frac{69,184}{5,165,433}$
$\mathrm{P}(\mathrm{2\ of\ 6}) = \frac{\binom{6}{2} \binom{48}{4}}{\binom{54}{6}} = \frac{2,918,700}{25,827,165} = \frac{64,860}{573,937}$

So my friends play the base game so they only spend \$1 per ticket. If we want to calculate the expected earnings per ticket they buy (minus the \$1 they spend), we need to take the probabilities we calculated above and multiply each of them by their respective prize amount (the 2nd column of the table).

$\mathrm{E(ticket)} = 11,999,999(\frac{1}{25,827,165}) + 2,083(\frac{32}{2,869,685}) + 53(\frac{376}{573,937}) + 2(\frac{69,184}{5,165,433}) \approx 0.55$

Yikes, that’s an expected earnings of 55 cents per dollar spent. Not much of an earnings, more like a loss. So over time, my friends can expect to loose almost half of the money they spend playing Lotto Texas.

If they played the Extra! version of the game, their expected earnings (minus the \$2 they’d spend) per ticket bought would be below:

$\mathrm{E(Extra!\ ticket)} = 11,999,998(\frac{1}{25,827,165}) + 12,082(\frac{32}{2,869,685}) + 152(\frac{376}{573,937}) + 11(\frac{69,184}{5,165,433}) + 0(\frac{64,860}{573,937}) \approx 0.85$

That’s a better expected value if you play Extra! but you’re still loosing money over time. And these calculations are based upon the assumption that you don’t have to split any of the prize money because if you do, these expected value calculations only get worse.

Conclusion

Playing the Lotto isn’t a crime, but it’s best to know your odds of winning before you play. And if you still want to bet, then all power to you. But Lotto Texas is a sucker’s bet.