Monday, August 25, 2008

Always switch cases - Tonight on "Deal or No Deal" the overly-excitable contestant managed to get down to her chosen case and a single case on stage. One held $5000 and the other had the million-dollar prize; she chose "No Deal."

But before they opened the woman's briefcase, Howie offered her to switch her case with the one remaining unopened case. Foolishly, she stuck with her original case. But if she understood the Monty Hall problem, she would have known that her original case had only a one-in-26 chance of having the million dollars while the case on stage had an excellent 24-in-25 probability of having the top prize.

Hard to believe? Nope, simple math.

The final word - Dave at Hedgehog Report: "I hope to some day be in a position in my life where I could turn down a guaranteed $530,000 for at best a 50/50 chance at $1,000,000."

7 comments:

Jody said...

Deal or no deal is not the same as the Monty Hall problem.

If Howie opened up every case but one and offered you the switch, you take it. But since you randomly lucked your way to a situation where one of two cases has the million, you only have a 1/2 chance no matter which case you choose.

Wiki has my back.
This problem appears similar to the television show Deal or No Deal, however with each selection the Deal or No Deal player is just as likely to open the winning box as a losing one. Monty on the other hand, knows the contents and is forbidden from revealing the winner. Assuming the grand prize is still left with two boxes remaining, the Deal or No Deal player has a 50/50 chance that the initially selected box contains the grand prize.

Eric said...

Stupid Wikipedia, contradicting me!

Well, the way I look at it is: her case had a one in 26 chance (1/26) while all the rest had a (25/26) chance = 100%. But when she opened 24 cases, she winnowed the pool down so that the big prize was almost certainly in the big, original pool.

I think.

Wink said...

For further reading, Milton Friedman did some seminal work on locating big bucks and avoiding the Whammy.

a said...

no no no all got it wrong. you guys have to account for variable change.
she had 5 one million dollar cases making it a 5 out of 26 (19.23%) chance of picking it and 80.73% chance of picking anything else (19of26). So at the end when they are only 2 cases(5,000 and 1 million) left you are ask if you want to switch. 99% of society will think its 50/50. Its not its the same as the beginning 20/80.
you are most likely to have pick anything but the 1 million 19/26
so if you case is wrong. so when there is a choice of 2 case your case has a 20% of the million and switching the case with have a million dollars 80% of the time

you chose then switch at the
a cases at the end. RESULTS
1 .01 1 million
2 1 1 million
3 5 1 million
4 10 1 million
5 25 1 million
6 50 1 million
7 75 1 million
8 100 1 million
9 200 1 million
10 300 1 million
11 400 1 million
12 500 1 million
13 750 1 million
14 1000 1 million
15 5,000 1 million
16 10,000 1 million
17 25,000 1 million
18 50,000 1 million
19 75,000 1 million
20 100,000 1 million
21 200,000 1 million
22 1 million - 5,000
23 1 million - 5,000
24 1 million - 5,000
25 1 million - 5,000
26 1 million - 5,000

this scenario is also in the movie 21. so you are more likely to pick the case with out the million resulting in a 80% chance that you will get the million dollars if you switch.

"you always switch"
its only a 50/50 at the end if
you started with 13 one million dollar cases

Octavarium64 said...

Hopefully things will be set straight again. The two possible situations that cause the million to remain in the final two are when it was originally picked, and when neither the original pick nor the next 24 turn it up. Both of these have the same probability.

This applies to any number of $1,000,000 cases, for example five as here. The two possible situations that cause a single million to remain in the final two are when it was originally picked, and when the original pick and the next 24 select only four $1,000,000 cases. These are both 5/26 chances.

If both of them are millions, then the keep and swap cancel each other out, so nothing changes.

Anonymous said...

Here is an easy way to think about it...assuming that only 1 case has the top prize.

You get to pick 1 case out of 26. So choosing the million dollar case has odds of 1/26. Pretty straight forward.

Down to 2 cases with the option to switch and one has the million. Stay with your case and gamble that your 1/26 odds happened to work for you. Switch cases and your odds change to 1/2.

If you get to this point. Always switch!!

Anonymous said...

Actually if u do an internet search, u'll discover that the Monty Hall (forced elimination) theory to always switch cases doesn't apply to a random elimination game like deal or deal (where it makes no difference whether u switch or not).