Research, analysis, evaluation

Against the Odds

Everybody gambles these days, but not everyone understands how to gamble. This week, therefore, I thought I'd address the idea of playing the odds. A little statistical analysis clarifies this idea greatly. Of course, this article is not an inducement to wager or to break any laws. In fact, I hope it deters people from gambling.

One thing many people don't understand is what odds are. Formally they are the ratio of the potential winnings on a bet to the wager. So if you bet $20 on a winning horse at odds of exactly 3 to 1 (often written as 3-1), you get back your $20 plus winnings of 3 times $20, or $60. That's a total of $80.

The important thing about odds is that you can convert them to a percentage that is the same as the percentage of bets you need to win to break even at those odds. For example, if you make 100 bets of $20 at even money (that is, odds of 1-1), you get back $40 if you win and nothing if you lose. You'll be betting $2,000, so you have to win 50 bets to break even.

You can find the percentage you need to break even at any odds simply by dividing the second number in the odds (that is, the number following "to") by the total of the two numbers and multiplying by 100. For example, at odds of 3 to 1 you divide 1 by (3+1) to get .25, then multiply by 100 to get 25% you have to win 25% of your bets at 3-1 to break even. At odds of 9 to 2 you divide 2 by (9+2) to get .18, then multiply by 100 to get 18% you have to win 18% of your bets at 9-2 to break even.

Well, you want to do better than break even, so of course you have to win more than that percentage. Calculating this percentage will often alert you to the difficulties facing you, and it will also often alert you to really bad odds.

For example, let's suppose you want to bet that your team can win a best-of-seven series in four games. Let's further suppose that you think your team has a 60% chance of winning any game against its opponent. Consequently, the probability that it will win four games in a row is 60% X 60% X 60% X 60% = 13%.

That is, on the average you can expect to win 13 of every 100 bets like that, if your estimate of your team's ability is correct. You have to get odds of better than 7 to 1 to make even a little money, assuming that your estimate of your team's ability is correct in the first place. If you allow, as you should, for error in your assessment of your team's ability, you'll only accept higher odds.

So let's further suppose the teams are actually equal in ability. Then you need to get odds of 15 to 1 to break even. The last time I looked into the local sports lottery they were paying off at 7 to 1 for a winning four-game parlay. A well-run sports book should be able to make money on this type of bet at odds not much lower than 15 to 1, so the government's ability to get people to take 7 to 1 shows how deeply the Canadian public needs gambling education.

Anyway, to return to our theme, this type of analysis may seem too complicated for you. If it is, don't gamble. If you can't estimate either your chances of winning or your profit or loss from a series of bets you are only going to lose.


1. Sometimes at crap tables, for example odds are stated as 3 for 1, for example, rather than 3 to 1. That means they are being expressed as the ratio of the winnings plus wager to the original wager. If you bet $20 at 3 for 1 and win, you get back 3 times $20, or $60. That's the same as the total you'd get back at odds of 2 to 1.

2. At race tracks pre-race odds are approximate. For example, if the odds on a horse are shown as 3-1 that means they're at least 3 to 1 but less than 7 to 2.

Against the Odds © 2000, John FitzGerald
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