StephenLawrence - 12 December 2012 02:10 AM

I think we can see that if the pair is picked from finite options **2.** is always false as all cases will be essentially the same as my example. Do you think that is right?

Not necessarily so, as “all cases” in the TEP is **not the same** as that in your non-TEP example.

Your non-TEP example is equivalent to four envelopes in **two** existing groups fo two envelopes.

So, we have Group 1 ($20, $40) and group 2 ($20, $10).

With A as the amount in the selected envelope, A must be from either one of the two groups:

1. If A is the smaller amount, it could be from group 1 ($20) or from group 2 ($10).

2. If A is the larger amount, it could be from group 1 ($40) or from group 2 ($20).

So, **A varies in value** in both 1 and 2 depending on whether A is the smaller or the larger amount.

With A varying in values in both 1 and 2, there is **no specific A** to assign probabilities of A as either the smaller or the larger amount as in the TEP.

Thus, which A is the **reference A** to assign probability? Is it A ($20), A ($10), A ($40) or A ($20)?

With no reference A, it seems no probabilities can be determined at all.

However, as we can compare A to the other amount in any one group, the probability is 1/2 that A is either the smaller or the larger amount because there are four possible values of A of which two are the smaller and two are the larger amount.

So, step 2 is true even in the context of your non-TEP example.

OTOH, in the context of the TEP, Step 2 refer to A (denoted as the amount in the selected envelope) which **must be of a constant value and that is the reference A** but A (notwithstanding it is of a constant value) could be either the smaller or the larger amount as in either (A, 2A) or (A, 1/2A) but **not as variable value A** as in the above.

In the TEP, with **two and only two** mutually exclusive possible events of either (A, 2A) or (A, 1/2A) for any finite value of A, step 2 is valid as the probability that A is the smaller amount is 1/2 and that it is the larger amount is also 1/2, is clearly true.

So, the probability of what A is (as either the smaller or the larger amount) is 1/2 + 1/2 = 1, which is also obviously true as there are two and only two mutually exclusive possible events for any finite value of A.

OTOH, your non-TEP example is about **mutually inclusive events** as both groups 1 and 2 do exist at the same time. As such, step 2 does not necessarily apply to your example as step 2 is specific to the TEP whereby the events are mutually exclusive.

So, you are comparing the validity of Step 2 in the context of two entirely different games i.e. the TEP and your non-TEP which is inappropriate as it is like comparing apples and oranges.

Notwithstanding that, as I have shown above, step 2 is true even in the context of your non-TEP example.

Now imagine you look in the envelope and see that you have $20. What should you do? I’ll give the answer to save time, you should switch because you know the probability is 50/50.

In the TEP, yes, but in the context of your non-TEP example, it is not so simple.

Now let’s change it a bit. There are two pairs of envelopes one of which has been selected at random. You know that in each pair one envelope contains half the other. You don’t know what the pairs are though. In fact they are as in my previous example

($10, $20) ($20 $40)

You open your envelope and see you have $20. What should you do? (I think the answer to this is revealing)

With mutually inclusive events, as in your non-TEP example:

1. If $20 is selected from group 1 ($20, $40), by switching, gain is $20.

2. If $20 is selected from group 2 ($20, $10), by switching, loss is $10.

The probability of 1, P (1) =1/2 and the probability of 2, P (2) = 1/2.

And the probability of 1 and 2, P (1 and 2) = 2/4 = 1/2.

For mutually inclusive events, P (1 or 2) = P (1) + P (2) - P (1 and 2)

So, the probability of 1 or 2, P (1 or 2) = 1/2 + 1/2 - 1/2 = 1/2

As we don’t know whether we are in (1 or 2), we should consider the probability of 1 or 2 which is 1/2. We cannot consider the separate probabilities of 1 or 2 because we don’t know whether we are in 1 or 2.

This is not the same as in the TEP (with mutually exclusive events) whereby the probability of gain/loss is simply 1/2.

So, the benefit of switching is not so simple as in the TEP.

However, as the probability of 1 or 2 is 1/2, it is beneficial to switch.