Draft - partially organized - The Drunken Walk

Sample space

Assume the probability of having a boy or a girl is 50% each. The sample space for a person having two children is:

(girl, boy)(girl, girl)(boy, girl)(boy, girl)

Goat and car

A game show has three doors; behind two doors are goats and behind one door is a car. First the contestant chooses a door but does not open it. The host knows what is behind each door, and he then opens one of the two doors the contestant did not choose that has a goat behind it. The contestant can then choose whether to switch to the other unopened door.

Question: Should the contestant change their choice at this point?

Answer: Yes. From the sample space perspective, we should list all the scenarios, but there are many, so first let's see how many “types” of scenarios there are, then look at how many “individual” scenarios there are:

Scenario A: The door the contestant chose has a goat behind it. We know the host opened a door that also has a goat behind it, so the remaining unopened door must have the car behind it.

Scenario B: The door the contestant chose has the car behind it. Given the host opened a door with a goat behind it, the remaining unopened door must also have a goat behind it.

The proportion of case 1 is the same as the proportion where the door the contestant chose has a goat behind it, which is 2/3. In other words, in 2/3 of the scenarios, if we switch we will definitely win the car!

A girl named Florida