Puzzle2

Puzzle1 illustrated that if you know that your cousin has exactly two children and one of them is a girl, then the odds of the other being a boy is 2/3.

You still know that your cousin has exactly two children and one of them is a girl. But you also know that the girl's name is Myrtle. Now what are the odds the other is a boy?

It is no longer 67%. It is 50%!

First, let's list the birth order possibilities:

1 girl Myrtle -- boy
2 boy -- girl Myrtle
3 girl other -- boy
4 boy -- girl other
5 girl Myrtle -- girl other
7 girl other -- girl Myrtle
8 boy -- boy
9 girl Myrtle -- girl Myrtle

But 3, 4, and 8 are automatically ruled out. That leaves only five possibilities:

1 girl Myrtle -- boy
2 boy -- girl Myrtle
5 girl Myrtle -- girl not Myrtle
7 girl not Myrtle -- girl Myrtle
9 girl Myrtle -- girl Myrtle

But #9 is Unlikely enough to be ignored. So we are left with:

1 girl Myrtle -- boy
2 boy -- girl Myrtle
5 girl Myrtle -- girl other
7 girl other -- girl Myrtle

So the odds of the other being a boy is essentially 2/4, or 50%.

Now that you have that one firmly in your mind, try this problem:

You face three doors and are told that behind one of them is a new car. Behind the other two is a goat. You are asked to pick one. Obviously, the odds of picking the door with a new car is 33%.

One of the other doors is now opened; a goat stands behind it. You are given the choice to stay with your original pick or switch to the other unopened door.

Should you switch?

If you think it makes no difference, you are wrong. Google "The Monty Hall Problem."

One more problem:

Assume you know of no reason you should be HIV-positive. You take an HIV test anyway and it comes back positive. The test is accurate in 999 of 1,000 cases.

Should you be concerned? Are your chances of being infected really 99.9%?

Not really. Think of the population.

In a population of 10,000 people of your background, only 1 is infected with HIV. But of the others, 10 will test positive. Your chances of being OK are really 10 out of 11. Still of concern; get tested again!

Apply this to drug testing in athletes. The so-called "false positive" rate is useless unless some estimate of the relevant population is considered. The use of the false positive rate in criminal cases is known as "The Prosecutor's Fallacy" and has wrongly convicted many people.

For more on this, see pages 104 to 118 of THE DRUNKARD'S WALK by Leonard Mlodinow. Highly recommended.

John Burgeson

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