Stephen wrote:
> Shay wrote:
> 
>>
>> There are actually four cups. Two cups in the $300 total universe and 
>> two cups in the $150 total universe. No amount of switching will 
>> change the universe in which you exist and that universe was selected 
>> before you chose a $100 cup.
>>
>>  -Shay
> 
> Yes! You've got it. :-)
> 

Here's how the math really works out, as well as a better explanation of 
where Andrew got it wrong.

Andrew counted the built-in profit of the system as "winnings" he showed 
for switching.

Four cups, right?
{$50 | $100} ------ {$100 - $200}

I'll change the scenario just a bit to make it clearer. Contestant is 
allowed to select one set of two cups ($50-$100 and $100-200) and to 
then select one of the two cups from his selected set. His expectation 
from this game? ($50 + $100 + $100 + $200) / 4 = $112.50. That's what 
he's being *given* for playing the game, so that amount must be 
subtracted from his winnings to reveal the profits of his strategy.

So, subtract $112.50 from each of the four cups to get
{-$62.50 | -$12.50} ------ {-$12.50 | $87.5}

Now, back to Andrew's premise. The contestant turns over a cup to find 
-$12.50. Should he switch?
Possibility 1: He's in the low set[1] - expectation = -$37.50
Possibility 2: He's in the high set - expectation = +$37.50
A 50% probability either way.

That's the honest math.

  -Shay

[1] Actually, not *the* but *a* high set. There are two ways that could 
go, but it doesn't change the problem.

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