In the article above there is a serious error I think. Suppose we open an envelope and find $10 inside. According to the section "What if you open envelope A?" once you've opened an envelope, you should always switch. Furthermore it is argued that this decision would be empirically justified by the results we would find when carrying out the experiment repeatedly. This defies common sense. All we know is how much is in one envelope and that one envelope contains twice as much as the other. The probability of getting the larger amount by switching is 50% regardless of whether we have opened the envelope or not.
I can clarify what I mean in the following way. Suppose we carry out the experiment a hundred times each with two envelopes. Supposed each experiment is identical. We open one envelope in one experiment and find it has $10. There are two possible scenarios. Scenario A is that in every experiment one envelope contains $10 and the other contains $20. Scenario B is that in every experiment one envelope contains $10 and the other contains $5. If Scenario A is the actual experimental setup, we should always switch. If Scenario B is the actual setup, we should always stay. But we don't know if we're in Scenario A or Scenario B.
All we know is that one envelope contains twice as much as the other. I would suggest that we should apply the Principle of Indifference to Scenarios, assume Scenario A is just as probable as Scenario B. This would mean, as common sense suggests, that it makes no difference whether we switch envelopes or stick with the original envelope.
This argument does not solve the problem but it strongly suggests that the argument made in the "What if you open envelope A?" section must be incorrect. The claim that repeating the experiment repeatedly would show that switching is better than sticking only works if we're in Scenario A. But it fails in Scenario B. It is because we don't know which Scenario obtains that we should estimate the probability as 50 percent.
In the article above there is a serious error I think. Suppose we open an envelope and find $10 inside. According to the section "What if you open envelope A?" once you've opened an envelope, you should always switch. Furthermore it is argued that this decision would be empirically justified by the results we would find when carrying out the experiment repeatedly. This defies common sense. All we know is how much is in one envelope and that one envelope contains twice as much as the other. The probability of getting the larger amount by switching is 50% regardless of whether we have opened the envelope or not.
I can clarify what I mean in the following way. Suppose we carry out the experiment a hundred times each with two envelopes. Supposed each experiment is identical. We open one envelope in one experiment and find it has $10. There are two possible scenarios. Scenario A is that in every experiment one envelope contains $10 and the other contains $20. Scenario B is that in every experiment one envelope contains $10 and the other contains $5. If Scenario A is the actual experimental setup, we should always switch. If Scenario B is the actual setup, we should always stay. But we don't know if we're in Scenario A or Scenario B.
All we know is that one envelope contains twice as much as the other. I would suggest that we should apply the Principle of Indifference to Scenarios, assume Scenario A is just as probable as Scenario B. This would mean, as common sense suggests, that it makes no difference whether we switch envelopes or stick with the original envelope.
This argument does not solve the problem but it strongly suggests that the argument made in the "What if you open envelope A?" section must be incorrect. The claim that repeating the experiment repeatedly would show that switching is better than sticking only works if we're in Scenario A. But it fails in Scenario B. It is because we don't know which Scenario obtains that we should estimate the probability as 50 percent.
Thanks.