Don’t Expect the Expected Value

One day, over 50 years ago, we were visiting Lake Tahoe as a family, and dad went across the border to play keno. He came back elated: he had hit seven out of eight on one of his tickets, and won eleven hundred dollars. He proudly laid out fifty twenties and two fifties on the kitchen table. It was a magnificent sight.

The details of keno are unimportant here, except to note that keno is not a game of skill. Of course the house has an edge. In the long run, you will lose money playing keno no matter how you do it. Even my dad, who over the years has played a lot of keno, and won even bigger payouts, would probably admit that he might have a net lifetime loss.

So why do people play? There are lots of reasons, I’m sure, but one of them must be connected to that heartwarming anecdote: fifty years later, I remember the event clearly, as one of joy and wonder.

Let’s explore that using roulette, which is much simpler than keno. A roulette wheel has 18 red and 18 black numbered slots, plus a smaller number of green slots (often two). You can make many different bets, but we will stick with red and black. If you place a $1 bet on red, and it comes up red, you get $2 back (winning $1); if it comes up black or green, you lose your dollar.

This means that if you place 38 bets on red, you will, on average, win 18 of them. If each bet is $1, you will have paid out $38 and won 18 bets at $2 each, for a net loss of $2. You can reinterpret that as a rate, that is, for every $1.00 you bet, you will on average get back $36/38, or about 95 cents. This leads to the expected value: –$0.05 for each dollar bet.

But of course sometimes you win. Let’s use Fathom to make a simulation of 100 people playing roulette. Each of them will play 20 games at $1 per bet, and they will always bet on red. Their possible results will therefore range from –20 to +20 (in dollars). The graph shows what I got in one run of this simulation.

roulette 100 x 20
Net results from 100 players, 20 games each

In the graph, the 32 winners are selected. There are also 52 losers and 16 who broke even. The average among all 100 players is –1.14, or –0.057 per dollar, which fits with the expected-value calculation we made before.

But do you think those 32 winners care that the average was negative? Of course not! They are happy. They will tell their friends that they won at roulette.

Which leads us to the title of this piece: we often smugly say that the lottery is a tax on the statistically illiterate. But the simple, inescapable fact that the house wins overall—that the expected value is negative—ignores the fact that most of the time, you don’t get the expected value. Sometimes you get more, sometimes less. The simple expected value calculation assures the house that it will win, but it might be variability that propels people to play.

The simulation lets us explore further. What if you lost money overall, but during your play you had, at some time, been winning? Would you feel enticed, that you almost won? That if you played again, you could win? The next graph shows which players were “up,” that is, had a positive balance, at some point during their 20 plays at the wheel:

roulette 100 x 20 up
Net results from 100 players playing 20 games of roulette. The 74 blue squares represent players who were ahead (“up”) at some time during those 20 plays.

And there were 74 of them! Nearly three-quarters of the players arguably had reason to be excited and happy at some point during their play.

My dad is one of those blue squares. He will always go back and play more keno. Nowadays, in his nineties, he even participates in keno tournaments, the concept of which, as a stats educator, I find bizarre—but he enjoys them. When I asked him why he gambles when it’s set up for the house to win, he explained, first, that he always allocates a dollar amount that he is willing to lose and when that’s gone he always quits no matter what; and second, that he plays because he enjoys it, not to make money. It’s the possibility of winning that makes it fun, and worth the price. To which I say, more power to you.

Note to math teachers: You can imagine setting this up in many ways in your class, with and without technology. I start with students playing or simulating a number of plays (20 is good) and recording the data, so we see the distribution of results. Then we calculate the empirical average result, and figure out the empirical rate, thinking about how if you bet more, you win or lose more, but in proportion. Then we might do the theoretical calculation of the expected value, using natural frequencies, that is, thinking about 38 plays of roulette rather than working with decimals and percentages.

Finally, this “were you ever ahead” extension is a new idea for me, but I’m definitely going to use it next time I do this in a class. It’s trickier to capture in a computer simulation, but if you’re simulating manually, it’s relatively easy. Or you could just ask, “were you ever ‘up’?” and I bet students will remember.

I believe that one of our jobs as stats teachers should be to help inoculate students to the potential dangers of gambling addiction. So of course we talk about how the house really, truly, is set up to win, and you have no control over that. But it’s also essential in this lesson to use the data to reflect on why people gamble, what the pressures are, why you might hear more from winners than losers, and why you might think that you could win too.

And a comment about roulette versus the lottery (which is more like keno, by the way): Both roulette and the lottery have negative expected values, but the lottery is pernicious in a different way. Because the distribution is so skewed—you have fewer winners, but winners can win bigger—there is a tantalizing chance of being the one who wins millions. Do the small payouts serve as “blue squares,” encouraging more play? Probably. For a class, however, roulette is easier because you’re not simulating rare events.

Author: Tim Erickson

Math-science ed freelancer and sometime math and science teacher. Currently working on various projects.

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