Cool Envelopes

Probability Level 1

There are 3 envelopes, one with $1, another with $2, and the other with $4. You do not know which envelope has which amount of money.

After you pick one at random, you are told which envelope of the remaining 2 has the lowest value, but not the actual value of that envelope. What is the expected value if you switch to the last envelope remaining (the one that has not yet been mentioned)?

The answer is 3.33.

1 solution

Kyle Coughlin
Jul 11, 2015

Because there is a small number of cases, we can analyze each case individually.

Case 1: You hold the envelope with $n$ dollars. You will be told where the envelope with $2n$ dollars is, and switch to the envelope with $4n$ dollars.

Case 2: You hold the envelope with $2n$ dollars. You will be told where the envelope with $n$ dollars is, and switch to the envelope with $4n$ dollars.

Case 3: You hold the envelope with $4n$ dollars. You will be told where the envelope with $n$ dollars is, and switch to the envelope with $2n$ dollars.

Therefore, our expected outcome is the average of our outcomes, or $\frac{4+4+2}{3} = \boxed{\frac{10}{3}}$

I don't quite understand why n is used in the above solution...

Surely when you say the envelope with n dollars, you mean the one with $1 dollar?

Also, it'd be nice if the question was adjusted so I didn't have to type in a recurring decimal as an answer, as that confused me :)

Jeremy Ho - 5 years, 5 months ago

Nice! Could you now try it with 5 envelopes? After you pick one, you are told the 2 envelopes out of the 4 that have the lowest value. Assuming you pick randomly between picking either remaining envelope, what is the expected value?

Clive Chen - 5 years, 11 months ago

Poor wording to the problem. IF... "The one not yet mentioned" was: NOT the one you selected randomly and NOT the one designated as the lowest remaining:

The expected value is you'd end up with, as written, is 4, with a 66.7% likelihood. There is a 33.3% chance you'd wind up with 2 and 0 chance you would wind up with 1. Possible scenarios: You pick 1, are told 2 is the lowest remaining and so select 4. You pick 2, are told 1 is the lowest remaining and so select 4. You pick 4, are told 1 is the lowest remaining and so select 2 .

walt winslow - 3 years, 10 months ago

Cool Envelopes

The answer is 3.33.

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