2012 update: I had a chat to Chris Coleman of ABC Riverina about the maths behind two-up. Check it out here and read on for the 2009 article on the maths.
It's an Australian tradition on ANZAC Day to take yourself down to your local pub and play Two-up - an Aussie gambling game in which you toss two coins in the air and bet on the outcome.
I'm somewhat embarrassed to say that even though I am only a month away from turning 30, this year was the first time I've ever actually gambled on two-up.
It's not a game that is played very often, despite being iconically Australian - according to the GAMBLING (TWO-UP) ACT 1998, outside of casinos it is only legal to play two-up on commemorative days like ANZAC Day (unless you're in Broken Hill, where the local council can legally arrange a two-up game any day of the year).
The rules of two-up are pretty simple. The Spinner places two coins (traditionally pennies) on a small piece of wood (the kip) and tosses the coins into the air. In the version of two-up we played at the pub, the gambling was very simple. Players standing around the Spinner either gambled on HEADS - which is where both coins come up heads - or TAILS - which is where both coins come up tails. If a head and a tail come up, the coins are tossed again and no one wins or loses. To bet, you find someone else willing to gamble the same amount but opposite to you, and then you have a one-on-one contest. If you want to bet $10 on HEADS, then you find someone willing to bet $10 on TAILS, and if you win you get their $10 - if you lose, you hand over $10. It's very simple and I love its inbuilt honour system.
The probabilities involved are simple too - you have a 50% chance of winning each time you bet. At the start of our ANZAC day down in Balmain, most people were betting $5. By the end of the day, as more beers were consumed, many were betting $50 and $100. Gambler's Ruin also started to show it's head - many people think that by doubling your bet after you lose you can get yourself back into the game. This doesn't work in this form of two-up for a couple of reasons. The first is that you need to find someone willing to bet the same amount as you, which is increasingly unlikely the larger you want to bet. And secondly, unless you have unlimited funds (or strictly speaking, more than everyone else you could bet against - or the casino if you are gambling there), it is highly unlikely that you could continually bet without going out backwards.
Two-up is also played in casinos and other gambling houses, and not just on ANZAC day. The rules, as you would expect from such institutions, are not so simple. In this expanded form of the game, there are a number of ways to bet. The South Australian Government has a good guide to two-up play, but simply put:
Players can bet in the following ways:
1) HEADS - odds of 1/1 ($1 bet pays $2, including your original $1);
2) TAILS - odds of 1/1;
3) 5 consecutive ODDS - odds of 25/1 ($1 bet pays $26).
The Spinner can bet in the following ways:
1) 3 HEADS are thrown before TAILS is thrown and before 5 consecutive ODDS are thrown - odds of 7.5/1 ($1 bet pays $8.50);
2) 3 TAILS are thrown before HEADS is thrown and before 5 consecutive ODDS are thrown - odds of 7.5/1.
This makes the game a little bit more interesting. The Wizard of Odds website for two-up sets out the probabilities for each of these outcomes - let's derive where they come from. At each toss of the kip, for this analysis it is best to think of there being 3 possible outcomes - HEADS, TAILS or 5 consecutive ODDS. We think of it this way because if a single ODDS is thrown, it is re-thrown and only makes a difference if it is one of five in a row.
As you can see, the House is paying out as if the odds are better than they actually are. It's not much, but this is how they make their money.
Again we can see, the House is not paying enough for a win - the odds should be 7.8 to 1, rather than 7.5 to 1. However, were you to back HEADS on each throw rather than as the group of three, the house would offer you odds of 7 to 1 (this is left as an exercise for the reader...), so the spinner's bet is better.
As it turns out, I came out even at the end of the day! There's some more maths to be had here - sometime soon we might take a look at some of these pay-out distributions.