Many recreational games combine skill and chance. Skill you can work on, chance is, well, a matter of luck. For all the ‘games’ discussed here, it is easy to persuade yourself that there is some finite list of outcomes, all equally likely. Thus in this chapter, unless explicitly stated otherwise, we will use the classical approach to finding probabilities: count the number of possible outcomes, and the probability of any event is taken as the proportion of those outcomes where the event happens.
My aim is to show how the ideas of probability can help a player make good decisions under conditions of uncertainty. An understanding of probability can also add to the enjoyment or entertainment of spectators.
Lot teries
A common lottery format is that known as 6/49, as in the UK National Lottery. Here 49 rubber spheres, painted with different numbers, are whirled around a plastic tub, six are chosen at random. Gamblers pay ￡1 to select six numbers, and win a prize if their selection contains at least three of those winning numbers. But since only 50% of the takings go into the prize money, the mean return to Lottery players is far less than in casinos, or at the racetrack.
The main attraction is the prospect, however remote, of an enormous prize – one UK ticket has won over ￡22 million, and prizes in the USA have exceeded $300 million. Counting tells us that the probability of winning a share of the top prize from one ticket, by matching all the winning numbers, is about one in 14 million in the UK, less than one in 116 million in the Euromillions game, and around one in 176 million in the USA Mega Millions game.
To appreciate just how tiny these chances are, fix on the UK game. Figures show that the probability of death within one year for a random 40-year-old man is around one in a thousand. So the chance of his death within a day is about one in 365,000, within an hour it is about one in 9 million, so to get down to one in 14 million we are talking about the chance he dies within the next 35 minutes! For the Mega Millions game, under the same assumptions, the chance his ticket wins a Jackpot share is comparable to his chance of death within the next three minutes.