Cognitive Illusions Part 2 :

[Kramer]

When a contestant is presented with the original 26 cases, he or she has a 3.85% (1 in 26) chance of selecting a case containing any of the available dollar amounts. (Compare this to the standard American roulette wheel, where selecting any of the available 38 numbers offers casino players just 2.63% (1 in 38) of selecting a winning number.)

If the contestant were allowed to open this case immediately, theoretically it would contain $750 or less half the time (13 of 26 cases) and $1,000 or more half the time (13 of 26 cases); the median ("middle") case value is $875.

However, because of the very large top prizes, the mean ("average") value of that case is $131,477.54.

If every contestant refused every deal (eventually being able to open their initial choice), the gameshow would expect to pay out approximately $131,131 per contestant on average.

However, most of the gameshow's payouts would be concentrated in a few big winners -- and most contestants would leave with very disappointing earnings.

Once the contestant has revealed six cases, he or she has a 5% (1 in 20) chance of the case containing any of the remaining available amounts.

The mean and median expected contents of the initial case change accordingly.

After the contestant has revealed five more cases, he or she has a 6.67% (1 in 15) chance of the case containing any of the remaining available amounts.
After the contestant has revealed four more cases, he or she has a 9.1% (1 in 11) chance of the case containing any of the remaining available amounts.
After the contestant has revealed three more cases, he or she has a 12.5% (1 in 8) chance of the case containing any of the remaining available amounts.

After the contestant has revealed two more cases, he or she has a 16.7% (1 in 6) chance of the case containing any of the remaining available amounts.
The contestant's odds of his or her selected case containing a specific value will continue to increase (20% . 1 in 5; 25% . 1 in 4; 33% . 1 in 3) until just two cases (the first selected case and the last case held by a model) remain.

At this point, the odds of winning either amount is 50% (1 in 2), regardless of whether the player switches the cases or not.

NOW LETS LOOK AT THE DEALS :

THERE ARE 3 CASES LEFT (PLUS YOURS)
1 HAS $ 1,000,000
OTHER 3 ARE $ 100 , $ 1000 , $50

THERE’S A 25% chance your case has $ 1000000
a 75% chance it doesn’t.

The bank offers you $ 220,000

you have a 1/3 chance of knocking out the million

But are these odds correct ?

perhaps there is still a 96.15% chance the million dollars is not in your case but in
one of the other 3 ?

THOUGHTS?