I was recently listening to the album Johnny the Fox, recorded by the second most famous Irish rock band. The opening song on the album, “Johnny”, is a tale of a young man being pursued after having committed a crime. The climax of the song finds him in an alley, surrounded by police, and the song’s narrator describes his potential fate:
Five to one he gets away
That’s the odds I’m gonna give
Five to four they blast him away
Three to one he’s gonna live
But what exactly does that mean? Giving odds of 3 to 1 means that if someone were to bet that Johnny gets away, for each 1 dollar that was wagered, the house would pay winnings of 3 dollars. If I offer 3:1 odds, I must believe that for every 1 dollar winning bet, there will be a matching 3 dollars bet on losing outcomes. Thus 3 to 1 odds corresponds to the judgement that 1 out of every 4 dollars wagered will be on that particular outcome, a probability of 1/4.
In the case of the song “Johnny”, we find that the odds being offered (5:1, 5:4, and 3:1) correspond to the probabilities 1/6, 4/9, and 1/4, which add to (6+16+9)/36 = 31/36. Since these probabilities don’t add to 1, this isn’t a zero-sum game, and someone has the advantage. But who?
When it comes time to pay off the bets, the situation is quite clear. One of the potential outcomes has actually occurred, and the payoff odds (assuming a zero-sum game) will simply be the ratio of the total money wagered divided by the amount wagered on that outcome.
In the case of three possible outcomes, if we assume that the total wagers have been $a, $b, and $c on the three outcomes, and set $s = $a+$b+$c, we’d have odds of
with corresponding probabilities a/s, b/s, c/s, adding to 1.
Alternately, if the house takes a cut of the total wagers as profit (as happens at parimutuel betting facilities, such as race tracks, in the USA), the amount of money available to pay to gamblers isn’t $s, but rather $s-p, where $p is the profit from the race for the house. In that case, the odds offered would be
whose probabilities would be a/(s-p), b/(s-p), and c/(s-p), with sum s/(s-p), a probability that is slightly more than 100%.
In general, we’d expect the sum of the probabilities from the odds to add to more than 1, and it appears that the scenario described by lyricist Phil Lynott in “Johnny” isn’t one he will profit from. Good thing Phil was a musician and not a bookie.
In the first race on March 14 at Bay Meadows Race Track, the actual odds after the race were 21:10, 309:10, 3:5, 31:5, and 9:1; as probabilities, these become 10/31, 10/319, 5/8, 5/36, and 1/10, whose sum is 4335479 / 3560040 = 4335479 / (4335479 – 775439), so in practice they had set aside $755439 out of every $4,335,479 wagered (roughly 18%).
There are several notions of probability lurking here. The bookie setting odds is trying to anticipate how gamblers are going to wager. In theory, gamblers are trying to anticipate the actual likelihood of each outcome occuring, then comparing that to the odds offered, and maximizing their expected value. But the bookies know this, and act accordingly. And from there it gets complex.
Shades of: In theory, theory and practice are the same. In practice, they are quite different. (Paraphrasing Einstein, or Popper, or Yogi Berra, or any one of a number of 20th C philosophers….)