Probability
Paul Erdös was one of the most prolific mathematicians
of all time. He was an example of someone with completely lopsided development
and when things went wrong, or didn't work out as he would have liked, he blamed
God who he called the Supreme Fascist. He never went to school and at three
he could multiply three-digit numbers in his head and amused visitors by asking
their ages and computing how many seconds they had lived. At four he started
looking for patterns to prime numbers.
There was a childlike quality about him. He suffered from cataracts but was
reluctant to get them seen to until he found it difficult to read. Finally,
under pressure from friends he agreed but first gave the surgeon the third-degree.
‘Will I be able to read’ he asked. ‘Of course’ replied the surgeon. So on the
day of the operation the anesthetist came to give him a jab. ‘What does this
do?’ Erdös asked him. ‘It puts you to sleep for the operation’ replied the anesthetist
‘What!’ Erdös was extremely agitated, ‘I thought you said I would be able to
read!’.
Erdös however, was fooled by a problem posed by Marilyn vos Savant's in Parade Magazine. The problem is usually called the ‘Monty Hall Problem’ after the show
’Let's Make A Deal’ and goes like this:
Suppose you're on a 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, number 1 - and the host, who knows what's behind the doors, opens another door - say, number 3 - which has a goat. He then says to you, "Do you want to pick door number 2?" Is it your advantage to switch your choice?
Erdös thought as most people that it made no difference and that the probability of a winning door number 1 or door number 2 is the same.
Here's another one:
In a two-child family, one child is a boy. What is the probability that the other child is a girl?
What if the older child is a boy? Does this information change the probability that the second child is a girl?
The answers to these questions defy common sense, which makes me very wary when the Sceptics society talks about statistical probabilities on small number of tests, and repeats tests in an inconsistent way as I discussed previously (‘Deconstructing Scepticism’).
For those who want to know, the answers to the questions are as follows:
- Yes, you should switch. The first door has a 1/3 chance of winning the car, but the second has a 2/3 chance of winning the car. For details see here
- The probability of the second child being a girl is 2/3. For more details see here
