Monday, November 29, 2010

A conditional probability issue?

There is a question and answer from 'AskMarilyn' at I copy the question and answer here since it is a probability issue.

Question: Four identical sealed envelopes are on a table. One contains a $100 bill. You select an envelope at random and hold it in your hand without opening it. Two of the three remaining envelopes are then removed and set aside, still sealed. You are told that they are empty. You are not given the choice of keeping the envelope you selected or exchanging it for the one on the table. What should you do? A) Keep your envelope; B) switch it; or C) it doesn't matter.

Marilyn said you should switch envelopes. Here's her reason: Imagine playing this game repeatedly. You start with a 25% chance of choosing the envelope with the cash. Then two empty ones are taken away on purpose. (Only someone with knowledge of the contents can inform you that sealed envelopes are empty.) so if the $100 bill is in any of the three unchosen envelopes - which it is 75% of the time - you'll get it by switching.

However, I would choose the answer C) it doesn't matter. This is a conditional probability issue. In the beginning, with all four envelopes sealed, the probability of choosing one envelope with $100 bill is 25%. When two envelopes are revealed not to contain the $100 bill, for the remaining two envelopes, each now has 50% probability with $100 bill in it. It doesn't matter if you keep the envelope on hand or switch it for the one on the table.


Cody Custis said...

D) Write a computer simulator of the four door Monty Hall problem.

Anonymous said...

You definitely switch envelopes. This is one the classic questions given to stats 101 students.

Dan Scharf said...

Think of it this way: I'm going to partition the set of 4 envelopes into a subset of 1 and a subset of 3. You initially get the subset of 1. Do you want to switch? Removing two empty envelopes is just slight of hand.