## Monty Hall problem and Two envelopes problem

### August 29, 2009

Monty Hall problem.
This problem became popular after the famous american show, where a participant at random choose one of three doors. Behind one door there is a car, behind others there are goats. Host knows where a car is and after your choosing leave your door closed, then opens the door with a goat from one of two left doors. And now the question from the host to a player: “Would you change the door?”.

For example, you chose first door. The host opened second door and show you a goat, then asked you to stay with the first door or change to the third.

Probability says yes, you should change! There are many answers to this problem in the internet, but they are so complicated, I try to make it simple. Look at the probability tree:

If you choose strategy “Stay” you win 1 car and 2 goats.
If you choose strategy “Change” you win 2 cars and 1 goat.
So it is better to change, you raise your chances to win!

Two envelopes problem.
There are two envelopes with money. In one envelope the sum is as twice as the sum in another. You do not know where is the envelope with the twice sum. Then you choose at random an envelope and open it, see the sum. Would you take the second envelope instead of the opened? For example, you see \$10, now if you change you can get \$5 or \$20 in the second envelope or stay with \$10.

There appears the problem.

1. Consider the case when there are two sum of money \$10 and \$20 in the envelopes, look at the picture what is happening when you make decision:

As you can see there is no deference.

2. Consider another case were you see \$10 in the envelope, hence in the second envelope there may be \$5 or \$20:

Expected value is $P(stay)=1/2 * 10=5, ~~ P(change)=1/4 * 5+1/4 * 20 = 7.5.$
So in this example it is better to change and your gain will be 1.5 times more with change strategy. The secret is that if you change you can lose \$5 or add \$10, what is not the same.

Simple program on C# shows next results for two considered methods (at every play there are random money in the envelopes):

In this simulation ultimate sums are approximately equal in the first method, in the second method if you change envelope you will raise the sum.