A bucketful of two-up

by Burkard Polster and Marty Ross

The Age, 27 April 2009

Anzac Day of course brought with it the traditional game of two-up. We have been pondering this, and we got sidetracked. The result is our new game: bucket-up.

For the diggers, with limited resources and much more pressing concerns, two-up had to be simple: two coins were thrown, and bets were placed. Standard bets were “Heads” or “Tails”, where both heads or both tails came up, respectively.

The probability of a head for each coin was 1/2, so the probabilities of Heads was 1/2 x 1/2 = 1/4, and similarly for Tails. The remaining possibility, of one head and one tail, was known as “Odds”, and had a 1/2 chance of occurring. Traditionally, Odds was not bet upon, and simply meant “toss again”.

But if Odds didn’t count, this left Heads and Tails as the equally likely outcomes. So, why not just flip a single coin? Possibly because using two coins made it harder to cheat. Or possibly two coins made the game more entertaining, by adding a delaying tension when Odds occurred.

Whatever the reason, if two-up is better than a single coin, then four-up must be better still. But, with four coins or more, you have to be careful.

There are many ways four coins can come up. What about Odds, where you bet that exactly two heads and two tails will occur. Would you take that as a fair 50-50 bet?

Don’t fall for this. In fact, it is a well-known con. There are 16 ways in which the coins can come up. And of these 16 ways, only 6 give equal heads and tails. So the probability of Odds is 6/16 = 3/8.

The chances of Heads or Tails in four-up are much smaller, only 1/16 each. The more coins we use, the smaller are the chances that Heads and Tails and Odds occur. But the way these chances diminish is paradoxical.

For example, how many coins do you think are needed so that Odds is less likely than winning the big prize in Tattslotto? We’ll give you two clues. First, 30 coins is enough to ensure that Heads is less likely than winning Tattslotto. Second, count the number of grains of sand that fit in an olympic-sized swimming pool.

We now give you the logical extreme of two-up: the game of bucket-up. What you do is take a bucket of coins and throw them up in the air. Then, the chances that an even number of heads turn up, or that an odd number of heads turn up, are both exactly 1/2.

The fact that bucket-up is truly a 50-50 game is not obvious. Contact us if you’re curious. More importantly, is bucket-up really an improvement over two-up? For the definitive answer, gather some children: then ask them whether they would prefer to play two-up or bucket-up.