This page provides experiment about the probabilities which are concerned on the two envelopes problem.
I hope it help you to realize the characteristics of these probabilities.
In brief the two envelopes problem is a paradox which is derived from an expectation calculation.
An unexpected amount of money is placed in one envelope, and twice that amount is placed in another envelope.
You chose one envelope at random.
Let x be the amount of money in your envelope, then the expected amount of money in the other envelope is (1/2)(x/2) + (1/2)(2x) = (5/4)x > x.
This means that the other envelope is more favorable than the chosen envelope.
(← Added on January 21, 2018.)
This calculation does not depend on the amount of money in your envelope.
Therefore when you choose an envelope the other envelope is always more favorable even if you do not know both amounts of money.
This is a paradox.!!!.
This experiment may help to realize the following facts.
The probabilities which are used to calculate expected amount of money of the other envelope are not necessarily 1/2. In other words the expectation formula should be corrected from "(1/2)x/2 + (1/2)2x" to "p (x/2) + (1 - p) 2x".
The equivalence of the two envelopes as a whole are held.
Even if the other envelope is more favorable for an amount of money of the chosen envelope.
If you are interested in it,
Experiment of the two envelopes problem
In this experiment two pairs of amount of money are used.
One is a pair of 200 and 400. The another is a pair of 400 and 800.
Select ratio of the probabilities of these pairs. Then experiment starts.
↓ Experiment starts
← Select ratio of the probabilities of each pair
Experiment started
←
chose one envelope
Pair of amounts
(200,400)
(400,800)
Ratios of probability of the pairs
Chosen amount
200
400
400
800
The Other amount
400
200
800
400
Count of occurrence
Ratios of the count of occurrence of the pairs
proportion of occurrences under the condition that 400 is chosen
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–
Average of the other amount under the condition that 400 is chosen
–
–
average of the chosen amount
average of the other amount
Some possible findings from these experiments
Terms
In the below I use the following terms.
"Other-is-Half" denotes the event that the amount of money of the other envelope is half the amount of money of the chosen envelope.
"Other-is-Double" denotes the event that the amount of money of the other envelope is double the amount of money of the chosen envelope.
"400-is-Chosen" denotes the event that the amount of money of the chosen envelope is 400.
Finding about conditional probabilities
The ratio of the following conditional probabilities equals to the ratio of the probabilities of the pair (200, 400) and the pair (400, 800).
conditional probability of Other-is-Half which is conditioned on 400-is-Chosen
conditional probability of Other-is-Double which is conditioned on 400-is-Chosen
Example of result of experiment
Finding about avarage amount of money
The equivalence of the two envelopes as a whole are held.
Even if the other envelope is more favorable or less favorable for an amount of money of the chosen envelope.
Example of result of experiment
Finding about expected value of amount of money in the other envelope
The expected value of amount of money of the other envelope depends on the ratio of probability of the two pairs.
Especially if the ratio is 1 to 1 then the expected value is (5/4) times the chosen amount.
And if the ratio is 2 to 1 then the expected value equals to the chosen amount.
