December 2010 Archives

I was trying to explain the Monty Hall problem to Alyssa the other night and failing miserably.

Imagine that the set of Monty Hall's game show Let's Make a Deal has three closed doors. Behind one of these doors is a car; behind the other two are goats. The contestant does not know where the car is, but Monty Hall does.

The contestant picks a door and Monty opens one of the remaining doors, one he knows doesn't hide the car. If the contestant has already chosen the correct door, Monty is equally likely to open either of the two remaining doors.

After Monty has shown a goat behind the door that he opens, the contestant is always given the option to switch doors. What is the probability of winning the car if she stays with her first choice? What if she decides to switch?

The correct answer is the contestant should always change their mind, by doing so they increase the odds of winning from 1 in 3 to 2 in 3.  This is pretty counterintuitive, and even experts get it wrong.  But it makes more sense if you draw it out:

Keep your choice:

C G G Initial Config
X     Your choice
  H   Host's Choice (he could choose either goat w/same results)
X     Keep your choice (WIN)

C G G Initial Config
  X   Your choice
    H Host's Choice
  X   Keep your choice (LOSE)

C G G Initial Config
    X Your choice
  H   Hosts Choice
    X Keep your choice (LOSE)

Change your choice:
C G G Initial Config
X     Your choice
  H   Hosts Choice (he could choose either goat w/same results)
    X Change your choice (LOSE)

C G G Initial Config
  X    Your choice
    H  Hosts Choice
X     Change your choice (WIN)

C G G Initial Config
    X Your choice
  H   Hosts Choice
X     Change your choice (WIN)