June Trivia: Answers to the Monty Hall problem

The Monty Hall problem

"Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you: 'Do you want to pick door #2?' Is it to your advantage to switch your choice of doors?"

In response to the Monty Hall problem, Marilyn vos Savant replied, that it would always be in the best interest of the contestant to switch doors as this would yield a 2/3 chance of success whereas not changing doors would only yield a 1/3 chance of success.

This answer triggered responses from thousands of readers, nearly all arguing that both doors stand an equal chance of success. A follow-up article reaffirming her anwer intensified the debate and some even questioned the integrity of her intelligence. However, careful analysis of vos Savant's answer actually shows that her answer is indeed accurate, as many math teachers will testify because it has now become a textbook example for all math topics on probability.

As seen in the diagram, 2 out of 3 times, the contestant will win a car if he/she switches doors. This is based on the assumption of course that the host always reveals a door showing a goat.

"Two Boys" problem

"Say that a woman and a man (who are unrelated) each has two children. We know that at least one of the woman's children is a boy and that the man's oldest child is a boy. Can you explain why the chances that the woman has two boys do not equal the chances that the man has two boys?"

As for the "Two Boys" problem, vos Savant said that the odds of the woman having 2 boys were 1 out of 3 while that of the man was 1 out of 2. Once again, readers argued that the probability should be 1 out of 2 in both cases. And once again, readers were proven wrong. Here's why,

To prove her point, vos Savant started a survey, calling on women readers (with exactly two children and at least one boy) and male readers (with exactly two children - the elder a boy) to tell her the sex of both children. With almost eighteen thousand responses, the results showed 35.9% (a little over 1 in 3) with two boys.