What’s the point of gambling?

In the BBC Radio 4 programme The Infinite Monkey Cage, Alex Bellos, author of the outstanding Alex’s Adventures in Numberland, was discussing randomness and made the following claim. 'I've interviewed lots of mathematicians and none of them say, "yeah I love gambling." They just don’t do it. I mean, what's the point?' I don’t know if I count as a mathematician, but I've made more trips to Las Vegas than I can remember. Were Alex interviewing me, I guess he’d ask me, "why?"

At school I studied Romeo and Juliet. I've been to see it in the theatre many times. Yet I know how it ends, so why do I bother going again? Now I do understand that, in the long run, a regular gambler will lose money. The casinos in Las Vegas are breathtakingly opulent and cost billions of dollars to build. That money came from the cumulative losses of the millions of visitors who take their chances there every year. Las Vegas is testament to the success of probability theory.

But this is to miss the point. Psychologists will tell you that the most addictive behaviours are triggered by regular but unpredictable rewards, which is exactly what you get from gambling. And sometimes those rewards are pretty big. On my last trip to Vegas I lost all my day's gambling money at the blackjack tables in the Bellagio. On my way out I passed a slot machine: I checked my pockets and found fifty cents. I put it in the machine and won. I then won some more. I soon had enough to return to the blackjack table, where the minimum bet was $10. I got lucky there, too, and within a few minutes I'd turned my initial 50¢ into $100 and I took a photograph of the chip to prove it!

Where else in life can you multiply your initial investment two hundredfold in a few minutes? Of course, that fantastic gain came after a very bad loss and at most I broke even. But it was fun! And I have yet another Vegas war story to tell.

Let me try to be a little more mathematical. One of my favourite games is craps, not least because it is rich in interesting probabilities and those probabilities are reasonably straightforward to calculate. (Roulette is a fairly dull game, and blackjack is a very complicated game.) If you place the basic craps bet, the "pass line" bet, the casino has a 1.4% advantage over you. That is, for every dollar you bet, the casino can expect to win about 1.4¢. Since the minimum bet is typically $10, you can expect to lose 14¢ per game. A typical craps game lasts for perhaps a minute, so you are likely to lose around $8 per hour. During that time you will receive one "free" drink.

OK, you've overpaid for one drink. But you've also had an hour's entertainment in a glamorous, exciting, sociable atmosphere. How much do you pay per hour to go to the theatre? Or the cinema? Or a nightclub? You might actually have won some money. Or you may have lost a lot of money. But if you're a regular gambler, your wins and losses will balance out to that average loss of just under ten dollars an hour.

But suppose you're not in it for the fun. You want to make money. How can you do that? Say you've got a thousand dollars. What should your strategy be? The answer depends on two things: how much risk you're willing to take and how much money you want to win. From a probability standpoint, the optimum strategy is to bet the whole lot in one go. If you split the money, you give the casino more chances to eat into it: remember every game is biased in their favour. If you place a thousand bets of one dollar it is virtually certain that you will come away with less money than you started with. If you place only one bet of a thousand dollars there is just over a fifty percent chance that you will lose money.

So what should that one bet be? If you want to win big, you can place one of the so-called "sucker" bets on the craps table. They're called sucker bets because the payout is poor relative to the risk. If you bet on "snake eyes" (rolling two dice and getting a one on both of them) the casino can expect to take your money at almost ten times the rate of the pass line bet. But on the other hand, if you do win you'll typically get around thirty times your stake. You could walk out with $30,000! The chances of this happening are 1 in 36. So it's very likely you'll leave with nothing. (And, if the bet were fairer, you'd be leaving with nearer $36,000, which is why you're a "sucker".)

If you're more risk-averse, you could stick with the pass line. You only stand to double your money, but the chances of you doing so are only very slightly less than 50:50.

Let me end with one more war story. On one visit to Vegas I found myself down $1,000. I had $500 left and desperately wanted to break even. I knew what I had to do: I bet the whole $500 on a single hand of blackjack -- and won. But I'd only recovered half my loss, so I had to do it again. Luck was on my side and I won the second hand. I ran from the table as fast as I could. The whole thing took about a minute.

Where else can you "earn" $1,000 in sixty seconds? Where else can you turn $1,000 into $30,000, if you're willing to risk losing it all?