StephenLawrence - 09 January 2012 11:28 PM

You’re viewing it as similar to this:

I have £10

On the toss of a coin I’m offered double or half so I should do it.

In the this case double or half is two different amounts £5 or £ 10 so I can gain more than I can lose.

Not so wrt the two envelopes.

Initially, you have nothing.

**You selected an envelope but you don’t know whether the envelope you selected has £10 or £20 or whether its amount is larger or smaller than the other envelope.**

Either:

You selected an envelope with £10 in it. On switching you have £20 in the other envelope

Or:

You selected an envelope with £20 in it. On switching you have £10 in the other envelope.

You end up with either £20 or £10.

Now, if you selected the £10 envelope and did not switch, you have missed the opportunity to get another £10. (100% gain)

OTOH, if you selected the £20 envelope and did not switch, you have avoided the loss of £10. (50% loss)

You end up with either £10 or £20.

However, the order of £10 and £20 is reversed wrt switching and not switching.

So, why switch if the final outcomes are £20 or £10, £10 or £20?

That might be the actuality for **one** instance, but what about **multiple** instances?

Because **you don’t know what is in both envelopes** and the odds are even, the guide to switch or not to switch is the expected value:

Where A is the amount in the selected envelope.

Whether A is £10 or £20, the expected value is more than A ....... you gain on average by switching.

If there were multiple instances of choosing the two envelopes, you gain on average by switching which means that there will be more £20 verses £10 outcomes and vice versa for not switching.

Paradoxically, this is recommended ad infinitum in steps 9, 10 and 11.