Should You Switch? Solving the 3 Doors Problem

[davee123]

Two new families move into your neighborhood. For the sake of argument, let's call one family Red, and the second family Blue. Each family has 2 children, whose genders you don't know. You're told that the Red's older child is a son. Then you ask the Blue parents if they have a son, to which they reply "yes". Which family is more likely to have a girl, and why?

DaveE

 

[vaishakh]

From the given information both equally.

 

[daveb]

Given that Red's oldest is a boy, the only possible combinationsa re boy-boy or boy-girl, a 1 in 2 chance. For Blue, the possibilites are boy-girl or girl-boy or boy-boy, or a 2 in 3 chance, so Blue has the higher probability of having a girl.

 

[davee123]

so Blue has the higher probability of having a girl.

Ding! It's a question of what information the person who told you has access to, or had to examine in order to tell you what they knew. In the first case, the person who told you that Red's oldest child was a son didn't need to know the gender of the younger child to answer you. For blue, they had to know both genders in order to correctly answer.

It creates an interesting quirk with the 3 doors problem:

You're on a game show, with 3 prize doors, one of which has a prize, one of which doesn't. You randomly pick the 1st door. Now, before revealing what's in the 1st door, the host opens up door #2, and shows you that there's no prize there. He then gives you an opportunity to switch. Should you?

The twist is that the problem (as stated here) is incomplete. It's unanswerable unless we know whether or not the host knew where the prize was. If the host doesn't know where the prize is, we're still at a 50% chance of our door being correct. If they host DOES know where the prize is, we're at a 33% chance of our door being correct.

DaveE