What are the odds? From coin tosses to card shuffles

If you’re playing a game of cards, Associate Professor Dale Roberts is a handy person to have around. He’s an expert on the mathematics of card shuffling, coin flips and games.

Roberts can tell you how many times you should shuffle your deck (seven) and how many coin tosses it’s likely to take to get four heads in row (30). But there’s much more to probability than quick calculations, and false theories can mess with our judgement.

“Probability theory has a long history — people have been studying it for hundreds of years,” Roberts says. “Even something as simple as flipping a coin can have a surprising amount of mathematical depth and plenty of real-world applications.”

Roberts says the game of two-up, traditionally played on Anzac Day, is a good example of how adding extra coins changes the odds. “For example, if you’re looking at the average number of flips to get four heads in a row, it takes around 30 flips for that to happen with one coin. With a two-coin game of two-up you’re looking at more like 340 flips for it to occur.”

A lot of players go into a game with a clear strategy but, according to Roberts, people can often be too eager to see patterns in the data. “People might think after a run of heads they’re overdue for a pair of tails, but that’s not how it works,” he says. “It’s not unlike a basketballer who starts hitting three-pointers and is said to have hot hands. But statistically with coin flips, each flip is independent of the last one.”

Probability can be applied to all sorts of scenarios. It can even help predict who someone is likely to vote for. “Mathematical models can help us understand different groups and how the individuals within those groups interact with each other,” Roberts says. “For example, if everyone around you votes Liberal, how likely is it to influence your vote?”

Mathematicians are always looking out for a phenomenon known as ‘phase transition’, which is like the physical process of water turning from liquid to solid. “For example, in a mathematically stylised voter model on a social network, just one additional connection in the interaction network can suddenly mean that a large amount of voters flip from red to blue,” Roberts explains.

We see this same phenomenon when shuffling cards. “There is a sharp transition from being barely random to very random at precisely seven riffle shuffles of a deck of cards.”

As compelling as streaks of luck and superstitions might be, buying into these theories will only stack the odds against you. Unless you can use the power of maths to pin down patterns in the data, Roberts says that when it comes to betting on a 50-50 outcome, such as heads or tails, “the best bet is to not bet at all”.