New Jersey Gov. Chris Christie is currently in a battle with sports leagues over the issue of allowing sports betting at casinos in Atlantic City and horse racing tracks across the state. If he wins and sports betting becomes legal in New Jersey, it will open the door for other states to follow suit. It appears there is a long way to go before this form of gambling spreads across the country.
But is sports betting really so much worse than casinos (which are legal in just under half of all U.S. states) or the lottery (which is legal in almost every U.S. state)? For the purposes of this discussion, we're going to ignore any moral and social issues and focus on just the statistics behind making each kind of bet.
If you had $10 burning a hole in your pocket, which form of gambling would be your best bet? I’m going to start by calculating the expected value for each one, and in subsequent posts we'll use Minitab Statistical Software and those expected values to see which type of bet is most risky.
Calculating Expected Values
An expected value is the amount that you’ll win “on average” on a single bet. For example, let’s bet on a coin flip. You bet $10 on tails (because tails never fails!). If it comes up tails you’ll profit by $10, otherwise you’ll lose $10. The probability of the coin coming up tails is 50%. So your expected value is:
(Odds of Winning)*(Profit) – (Odds of Losing)*(Amount Lost) = .5*$10 - .5*$10 = $5 - $5 = $0
On average, you won’t win or lose any money on this wager. Now we can apply this same formula to our three bets. I’ll start with the simplest, sports betting. Let’s say you bet the spread on a NFL game. I’ve previously found that betting on NFL games isn’t that different from betting on a coin flip, so I’m going to set the probability of winning the bet at 50%. However, the way the sportsbooks get you is that you’ll only profit $9.09 on your $10 bet. So that makes our expected value:
.5*$9.09 - .5*$10 = -$0.45
So on each $10 bet, you’ll lose about 45 cents. How will that compare to our other games? Let’s say that I want to try and win a little more money than just $9.09, so I walk into a casino and play a single number in roulette. With 38 different numbers, my probability of winning is 1/38 = 2.6%. Sounds low, but if my $10 bet wins, I’ll win $350! The other 97.4% of the time I’ll only lose $10. So what’s my expected value?
(1/38)*$350 – (37/38)*$10 = -$0.53
Despite the fact that I’ll win roulette much less frequently than my NFL bet, the payoff is so large that the expected value is about the same as football. But the value is still negative, so don’t think roulette is going to be a viable career path.
Least Controversial, But Most Expensive?
Let’s move on to the least controversial of these games of chance, the lottery. There are many different forms of the lottery, but to be consistent I’m going to pick a $10 scratch-off ticket called “Neon 9s” from my home state of Pennsylvania. The top prize is $300,000, with many other prizes ranging from $10 to $30,000. Since the odds for each prize are drastically different (you can find the complete list here), finding the expected value becomes much more complicated. But when you calculate it all out, you’ll find that the expected value of buying one ticket is:
Ouch! That’s much worse than the previous two expected values. You could make 6 sports bets or 5 roulette spins before you’d be expected to lose more money than buying one $10 scratch off ticket! The chance of winning that top prize may be alluring, but you sure pay a hefty price for that chance. If people lose so much more money (on average) playing the lottery, it makes you wonder why that form of gambling is "okay" and legal in almost every state, while the other two are often frowned upon and/or illegal in most states. Sending mixed messages, aren't we?
Anyway, it turns out that you'd lose your money least rapidly making a $10 sports bet. But this is all theoretical. Although the expected value is negative for each bet, people are still able to win when gambling. What would playing these games in the real world look like?
Say we take 300 different people and make them place a $10 bet every week for a year. One-third will make a $10 NFL wager, another third will bet $10 on a single roulette number, and the final third will buy the $10 Neon 9s lottery ticket.
Hmmm. I'm probably going to have a hard time convincing 300 of my friends, family and coworkers into joining this experiment. Luckily I can do it all in Minitab! So next week I'm going to come back and show you how to use Minitab to simulate making sports bets, playing roulette, and buying lottery tickets. Then I'm going to run my experiment, and see if anybody comes out ahead!