Should You Stay or Switch? Decoding the Probability in the Envelope Game Show

[Jameson]

Math Help Boards decides to host a game show with the prize being the member of your choice helping you with math for two hours over Skype... (Clapping)... Anyway, the set up is as follows:

There are four envelopes, one of them containing a ticket to redeem the tutoring prize and the other three contain nothing. I, the host, ask you to choose one of the envelopes and you do. Now to be tricky and try to get inside your head I open two of them, both of which are empty. Now I ask you do you want to stick with your original envelope that you chose or switch to the remaining envelope? This is a one time offer and once you decide you must immediately open the envelope you choose.

The question is what should you do: stay or switch, and why? What is the probability of opening the envelope with the prize if you stay or switch? Does it matter?

Note: The past few weeks have been much easier than in the past to try to mix things up a bit. Next week will be a much more challenging problem if you find this too easy.

Remember to read the http://www.mathhelpboards.com/threads/773-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Jameson]

Congratulations to the following members for their correct solutions:

1) Sudharaka

* Honorable mention goes to veronica1999.

Solution:

[sp] As many of you might already know this problem is a very slight variation of the famous Monty Hall Problem, which contains 3 doors instead of 4 envelopes but the principles of the analysis do not change.

The correct answer for our POTW is that you should always switch to the remaining envelope given the chance. The best non-rigorous way to understand and demonstrate why this answer is correct follows:

Note: There are two envelopes remaining when asked if you wish to switch - one envelope contains the prize and the other is empty. Thus, you cannot possibly switch from an empty envelope to an empty envelope or from the envelope with the prize to another envelope with the prize. Given this fact we know then that there are two possibilities when switching - correct to incorrect or incorrect to correct.

1) Correct to incorrect switch - this occurs when you picked the prize correctly at the beginning of the show but not knowing that of course you choose to switch envelopes and it doesn't work out well for you. How often does this happen though? In order for this to happen you must guess the correct envelope out of 4 choices, which will happen 1/4 of the time. Thus the probability of switching leading not getting the prize is 0.25 or 25%.

2) Incorrect to correct - this occurs when you did not pick the correct prize. How often does this happen? 3/4 times you will not guess the correct envelope. When this happens and you switch you are switching to the envelope with the prize since no other empty envelopes remain.

The Wikipedia page I linked to has lots of interesting comments on this problem such as why it is so hard for people to accept the correct solution! This simple problem demonstrates how poorly we intuitively understand probability and when conditional probability comes into play. The fact that the host is privy to information you are not and makes non random actions confuse many into thinking the chances are "fifty-fifty". [/sp]