Are you lucky? Can you change your luck? Do you carry any lucky things around with you?
GAMES OF CHANCE
People enjoy playing games of chance. They like luck to be a factor in the game. There is a branch of maths called Game Theory which explains the probability, chances or odds of winning games. Let’s look at some simple games of change.
I am sure you have tossed a coin. In England, the game gets called ‘Heads and Tails’ because on one side of English money is the Queen’s head, and on the other is a lion (they have tails). When we toss the coin, we call out heads or tails and see which side up the coin lands. Both heads and tails have equal probabilities and so the change of within is 50/50 or we can say 50% or 0.5 probability.
If we toss two coins at the same time, we get a more complicated game. The coins could be two heads, two tails or one head and one tail. What are the probabilities of these results? Well, remember that one head and one tail could be made two ways. A head and a tail are equal to a tail and a head. So there are really four possible results in the game but two are identical. So double heads and double tails have a 25% change each while one head and one tail together has a 50% chance.
Take three coins together and you change the odds again. We could have triple heads, triple tails, double heads and one tail plus double tails and one head. Can you work out the probabilities here? Well, triple heads are 12.5% and triple tails are 12.5% too. Double tails and one head gets 37.5% as does double heads and one tail.
MONTY HALL PROBLEM
The Monty Hall Show was a game show on American TV. It has become famous in giving its name to a mathematical game theory puzzle. This puzzle is rather confusing. It is counter-intuitive. That is, what you think happens is the opposite of what really happens.
In the game, the contestant is given three doors to choose. Behind two of the doors is a goat. Behind one door there is a car. The contestant first chooses one door. The presenter then opens one of the incorrect doors revealing a goat. The contestant is now given the chance to change their answer or stick with the original door. The door of choice is then opened and the contestant then either gets a goat or a car. Obviously, they want the car, not the goat.
Many people, when hearing the game, think it is fair. In the beginning, the probability of choosing the right door is 33.333% (1/3). One door is then revealed. The contestant is given the choice again. They probably think that since there are now two doors, that the probability of getting the right door is 50% (1/2) and so the game is fair. However, the truth is that they are not playing a fair game.
In reality, the probabilities of the two remaining doors are not equal. If you stick with the original choice, you will win 33.333% (1/3) of the time. If you change your answer then you will win 66.667% (2/3) of the time. Can you work out why?
If you don’t believe me, you can test it. Play the game 100 times and keep track of how often you win when using the sticking strategy. Then play another 100 times using the always change strategy. You will find the numbers work out correctly as 1/3 versus 2/3 and not the even 1/2 and 1/2 that you might have expected.