Neurobics

Odds Against

By Russell Dear

Assuming that I'm a mathematician and know about these things, people often ask me if I buy Lotto tickets, and if I use a formula to choose the numbers. My response is "I always pick the numbers 1,2,3,4,5 and 6" -- a reply guaranteed to garner disbelief, despite my insistence that these numbers stand as much chance of a win as any others.

Questions relating to probability are fraught with difficulty. People are quick to answer intuitively without thinking things through mathematically. In fact, the solutions to some problems are so counter-intuitive that people become quite heated in defence of their own views, even when incorrect. The following problem, from Parade, is a classic example: Suppose you're on a TV game show, and you're given a choice of three doors. Behind one door is a car; behind the others, goats. You pick a door -- say, No. 1 -- and the host, who knows what's behind the doors, opens another door to reveal a goat -- say, door No. 3. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choices?

The correct answer, that it is advantageous to switch choices, created immediate response. Letters poured in, and most (90%) disagreed. Interestingly, some of these letters were from university mathematicians.

One way of thinking about it is to realise that when you have chosen a door, the probability that there is a goat behind it is 2/3 (two of the three doors have a goat behind them), while the probability that it hides a car is 1/3.

You don't want to choose a goat, but it is twice as likely that you have done so. Without further information there would be no point in changing your choice, since you would still be more likely to choose a goat than the car.

If there is a goat behind your door, changing your choice gives you a 50% chance of winning the car. If the car is behind your door, changing your choice means you have no chance of winning it. However, once the game-show host has shown you where one of the goats is hidden, changing your choice improves the odds of your winning. Remember, it is twice as likely that a goat is behind your door. If a goat is there, changing your choice means you are certain to win the car.

I suspect that this type of conundrum stems from a question posed by Charles Dodgson in 1887, in his book Pillow Problems: a bag contains one counter, known to be either white or black. A white counter is put in, the bag shaken, and a counter drawn out, which proves to be white. What is the chance of now drawing a white counter?

Try this one. The Apple and Pear Marketing Board has a new way of deciding on managerial positions. It gives each hopeful contender 100 red apples, 100 green apples and three empty cartons. They are asked to place the apples in cartons in any way they like, so long as at least one apple is in each. Reds and greens may be mixed.

They are told that one carton will be chosen at random and one apple removed from it. If the apple is red, the hopeful contender gets the job. If the apple is green, they'll need to try again next year.

What should you, as a hopeful contender, do to maximise your chance of getting the job?

Russell Dear is a Mathematician living in Invercargill