Losing on a Winner

It was a feature of local news where I grew up that at the end of the sports report, a man in a tweed jacket would come on screen and recite a seemingly random sequence of names and numbers. I later worked out that he was the “tipster”, and that he was divulging his wisdom in the form of predictions as to who would win various horseraces the next day. Here is a more modern example.

I was thinking about this recently and realised something: every single horse racing “tip” is completely useless. Even if the tipster is almost completely certain that his tip will win the race, it doesn’t follow that you should bet on that horse.

Allow me to explain. Of course, if you are 100% certain that a given horse will win, then go ahead. But that will never be the case: even a rigged race may not go according to plan. So let’s suppose that you are 95% certain that a given horse will win. You should bet on that horse, right? It’s practically a sure thing.

Actually, no. You shouldn’t necessarily bet on that horse. Suppose that there are 20 races at the track today. And in each race you are 95% certain that you know who the winner will be. That means that, on average, you will be wrong for 5% of the 20 races, which is one race. On that race, you will lose the money you bet. So in order to come out ahead, you must make enough money from your 19 wins to make up for your one loss.

If the bookie gives you odds of $1.05 for each of the races, then (if you bet $1 on each of the 20 races) you can expect to go home with  $1.05*19 – $20 = -$0.05. It follows that you should not bet on those horses. On the other hand, if the bookie offers $1.10, you can expect to make a modest profit of $0.90. Generalising, if the probability of an outcome is p, and the bookie offers a price of $b (i.e. for every 1 dollar bet, you get back b dollars), then a good bet is one for which pb > 1.

The moral of the story is this: betting is not about who you think will win. It’s about getting good odds, which is where the bookie comes in. If the bookie has done his job right, he will ensure that no outcome has pb > 1. The easiest way to do this is to make b slightly less than 1/p. For example, where there are two teams who have an equal chance of winning (p = 0.5), bookies often offer odds of $1.85 on each. If you agree with the bookie that each team has an equal chance of winning, then you should not bet at all, since pb = 0.925 < 1 for both teams. So I’ll say it again: you only bet on an outcome when you think that the bookie has underestimated the probability of that outcome occurring. It follows that every betting tip ever given is completely useless, unless it takes into account the odds offered.

That said, I think that betting is one of the worst things that ever happened to sport. Nothing ruins a sporting contest faster than the suspicion that it has been rigged, and that suspicion will always be there as long as there is more money to be made in dishonesty than in honesty.