Two Up - a Game of Chance and a Study of Randomness

Two-up is a game of chance played on ANZAC day in Australia.

The odds of winning or losing are 50-50.

This instructable looks into the pattern of heads or tails and explains why doubling up to recover a loss is probably not a good idea.

A designated "spinner" throwing two coins or pennies into the air. Players bet on whether the coins will fall with both heads up or both tails up. If one coin lands heads and the other tails it doesn't count and the coins are thrown again.

Unlike games played in casino's there is no 'middle man' taking a cut.

My goal in playing 2 up was to have fun, drink some amber beverages and still be playing at the end.

Each year I set a limit I am prepaid to loss which I will call 'my float'. In past years this has been $50 for the entire afternoon. In some years this hadn't lasted very long.

I usually like to bet $5 a throw with the expectation that the odds and evens will cancel each other out and I will finish the afternoon with about the same money I had started with. This doesn't happened that often and I was interested to understand why.

A simulation of thowing Heads or Tails was run on a spreadsheet to see how often 'runs' of heads or tails occur. This is useful to know as it can determine the size of bet to play and float required to have a good chance of keeping playing for a long time. It was found a $50 starting float would mean you can probably keep playing if bets are limited to $5 but if bets are $10 the float will likely to be busted sometime during the afternoon. To have a good chance of playing the whole afternoon with $10 bets a higher float is required. Having a starting float is also a good way to set a losing limit.

In past years I noticed people tended to be betting $10 each throw. This year I increased my starting "float' to $100. I was prepared to loss this in exchange for a fun afternoon with a few drinks being extra.

So lets go over what the 4 photos shown.

Photo 1 : My 'float' I had with me, being $100 I was prepared to loss as the game progressed. I wanted to bet $5 a throw if possible but would accept $10 bets too. I also changed my bets while playing, i.e.didn't stick to Heads or Tails.

Photo 2: This is a spreadsheet simulation so hopefully will provide some insight into why when playing with a $50 float while betting $10 my float often gets busted. Using the excel RAND and IF statements, I simulated throwing Heads and Tails and the likely outcomes. To keep things simple I stuck to just Heads in the simulations. On this simulation my $50 float would have been busted on game 5 and my $100 float busted on game 20.

Photo 3: Another simulation, didn't get busted and finished $60 up after playing 64 games.

Photo 4: 20 simulated years on one screen (remember two up is only played one day a year). This photo shows the results of over 600 throws. It shows the Total for Heads and Tails in each simulated year after 64 games. (These varies by as much as 12 in a year) It also shows that after 1280 games there were only 6 more heads than Tails, pretty close to 50% Heads, 50% Tails. It also shows in one year, there was a run of 11 Heads.

Photo 1: Shows the simulation of 1260 games. for one person playing over a period of 20 years (again) and another table showing the total of 20 years tallies for 10 other players. As can be seen in total, Heads and Tails are almost equal. But for some of the players there is a difference of close to 60 in total, this would be $600 at $10 a bet. But what if each player had 3 beers during the afternoons, at $10 a beer? Well, each would have spent $30 every year, $600 over 20 years or $6,000 for the total 10 players. So, while the game is about even stevens in the long run, You will still be out of pocket on drinks and this ignores what else you could have been doing with your time (the opportunity cost).

Photo 2: Looks at having a $50 float every year, a $100 float or no float with 64 simulated throws each year. It shows that with a $50 float (maximin you are prepared to loss) if the games were $10 a throw (after 64 or less games) in 9 of the 20 years simulated, your float will have been busted (red numbers). In this simulation (playing Heads) the float wouldn't have been busted if it was set to $100. But look at the blue highlighted cells, if Tails only were played the $100 float would have been busted in 4 of the 20 years.
Conclusion: pIay at a place where people are making $5 bets if you only have a $50 float and your aim is to play for the whole afternoon. So much for looking at spreadsheets, lets see how the game actually played out.

A record of my bets placed was kept, if heads or tails were bet and if I lost or won. This was later entered into a spreadsheet to see any 'learnings could be obtained'.

So, what were the 'learnings'?

• 'Only bet what you are happy to lose' - an oldie but a goodie.
• 'Runs' happen but only in hindsight.
• 'Doubling up' can lose you a lot of money, if you do it to recover a loss and a run is against you. (While playing in the afternoon, I had a run of 8 loses, so if the doubling up method had been used to recover loses, on the 9th throw $1280 would theoretically needed to be bet, (i.e, 5 10 20 40 80 160 320 640 1280)
• Losing money hurts more than the pleasure of winning money but having a 'float' you are willing to lose seems to mitigate this to a degree. (I was down more than 80% of the time during the afternoon and was OK with it)
• Know why you are gambling and when to stop. (I always stop when my float is gone) In some years I doubled up to 'recover' on a lose, lost again and my float ended early in the afternoon, I still walked away.
• 'Simulations' help for a game plan but don't always play out in the game. (I had a longer run against me, and for me than the simulation would suggest)
• If my float had been set at $50 and I had always bet $10, it would have been busted on throw 13.
• My Record keeping may not have been 100% accurate as, it said I finished $50 up at the end, yet I only counted an extra $20. (Drinks were paid separately)
• Final telly of Heads and Tails was 33 Heads and 31 Tails. Petty close to even.
• 25 times the winning coin was the same as the previous one. I.e. Heads followed Heads or Tails followed Tails.
• If was a fun afternoon, making up part of Australia's Memorial Day.