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Lotto Texas: A Sucker’s Bet

Lotto Texas, and the lotto in general, is a sucker’s bet. Let me explain.

Millions of Americans play the lotto each year as an innocent way to gamble their savings on winning some money. If they lose, who cares, it helps to fund government programs like schools.

Or at least that’s what you think. Some studies coupled with an excellent report by John Oliver show that States actually anticipate lottery earnings. They don’t add that extra money to the base budget for programs like schools. Instead, they replace the base budget with lotto money while routing the rest of the budget towards other Stately concerns.

But this post is not about public policy. Rather, it’s 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 (at the time of writing)

The rules of Lotto Texas are quite simple:

If you play the base game, you spend 1 dollar per ticket. If you play the Extra! game, you pay 2 dollars 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 the lotto case, you’re trying to find the expected value of a ticket. That is, if you played the lotto enough times and calculated your earnings and losses, what would be the average value of each ticket you invested in?

For example, if you buy a two dollar ticket, but the expected value of that ticket is only one dollar, 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?

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

$\mathrm{P}(\mathrm{5\ of\ 6}) = \frac{\binom{6}{5} \binom{48}{1}}{\binom{54}{6}} = \frac{288}{25,827,165} = \frac{32}{2,829,685}$

$\mathrm{P}(\mathrm{4\ of\ 6}) = \frac{\binom{6}{4} \binom{48}{2}}{\binom{54}{6}} = \frac{16,920}{25,827,165} = \frac{376}{573,937}$

$\mathrm{P}(\mathrm{3\ of\ 6}) = \frac{\binom{6}{3} \binom{48}{3}}{\binom{54}{6}} = \frac{345,920}{25,827,165} = \frac{69,184}{5,165,433}$

$\mathrm{P}(\mathrm{2\ of\ 6}) = \frac{\binom{6}{2} \binom{48}{4}}{\binom{54}{6}} = \frac{2,918,700}{25,827,165} = \frac{64,860}{573,937}$

Since my friends play the base game, they only spend one dollar per ticket. If you want to calculate the expected earnings per ticket they buy (minus the one dollar they spend), you need to take the probabilities you 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 they 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 Lotto Texas 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.

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