I will suggest how probability ideas can help in making decisions in the face of uncertainty, and also describe circumstances where misunderstandings can arise.
Odds?
Recall that probabilities can be expressed in terms of odds, and vice versa: a probability of 1/5 is the same as odds of 4 to 1 against. Unfortunately, the term ‘odds’ has also been usurped by the gambling community to mean something quite different – the amount the bookies will pay if your selected horse wins. So when you read that Sea The Stars won the 2009 Derby at odds of 11 to 4, that simply means that for each ￡4 staked on the horse, the profit, because it won, is ￡11. The figures ‘11 to 4’ have no automatic relationship with the probability of winning. They depend on the bookies’ subjective assessments of the horse’s chances, and on how much money gamblers have staked. The term ‘payout price’ is a more accurate use of language for these figures like 11 to 4, but, regrettably, we have to accept the common usage of ‘odds’ in this gambling context.
A payout price is termed fair if it gives no monetary advantage to either party, i.e. the mean value of the gamble is zero. The fair payout price for correctly picking the suit of a card selected from a well-shuffled deck is 3 to 1, as those are the exact odds against a correct guess.
Commercial gambles are not fair, in this sense, as they could never operate without a house advantage. For roulette in a UK casino, when all 37 outcomes are equally likely, the payout price for betting on a single number is only 35 to 1, not the 36 to 1 that would be fair. So the mean return on a bet of ￡37 is ￡36, giving a house advantage – the percentage of any bet it expects to win – of 1/37, about 2.7%.
This advantage is the same for most of the available bets in roulette: whether you are betting on pairs of numbers, triples, groups of four, or six, or twelve, for every ￡37 you stake, your mean return is always ￡36. But in Las Vegas, the standard house advantage is bigger, because of an additional slot, double zero, giving 38 outcomes – with the same payout prices as in the UK. The mean return on $38 is generally $36, a house advantage of 2/38, or 5.3%.