sandsynlighed, please help!

I'm discussing this with some not-Danish friends, so please answer in English or if you can, in German.

The game is DEAL NO DEAL

There is 4 suit cases left, 3 on the stage, 1 in the hand of the contestant. She can only pick one of the 3 on the stage as her next suit case.

in the suit cases are: 100k€, 50k€, 10€ and 0,01€ . bank offer=45k€

What is now the probability that the contestant will pick a suit case under the bank offer?

50 %. in English :-)

Nå. har du vundet?

You only have 3 suit cases to choose from and you dont know what isin them, and therefore you cant decide what the probability is.

please, if this statement is'nt true, explain why it is'nt true.

Depends on the contestant.
If the player is a poker player he/she would probably realize the bank offer has higher EV than the value of the suitcase he/she had already chosen, and therefore the probability would be 0%, as he/she would maximize value, and take the bank offer.

If the player was random and chose a suitcase, and the boxes were fairly distributed, it would be 50%.

@akropolice
I hope you're not planning poker as a career choice :-)

Probability is all about quantifying the unknown based on available information. You are looking at the problem from a wrong perspective. Indeed we don't know what is in each of the *four* suitcases, but since we do know that two of them contain amounts larger than the bank's offer and two of them smaller amounts, 2/4 =50% of the times whichever suitcase we examine it will have a smaller amount.

If the 100K¤ = 1, 50K¤ = 2, 10¤ = 3 and 0,01¤ = 4 the amounts can be distrubuted among the 4 suitcases, A, B, C & D in the following ways:

ABCD
1234
1243
1324
1342
1423
1432
2134
2143
2314
2341
2413
2431
3124
3142
3214
3241
3412
3421
4123
4132
4231
4213
4312
4321

As you can see, each of the 4 prizes would be in a specific suitcase 6/24 times, or 25%. Therefore 50% of the time you would pick a prize under the bankoffer, and 50% of the time over.

Two suitcases over 45k¤ and to under, therefore 50%.

(100k + 50k + 10 + 0.01) / 4 = 40k.
The bank offers 45k > take the deal.

@ BingoBango
Yo yo yo, who said I think it's anything else than 50% ?
I'm just trying to keep things from an objective point of view.

My friends, however, still is'nt convinced, and it bothers me because i like to win these kind of arguments..

Can someone please try to explain it in a very understandable way?

(thx for the answers until this moment)

@Marryat


NO DEAL

pot odds m8

As BingoBango, clearly 2/4, I really don't see the issue.

2 suitcases have more than 46k
2 suitcases have less than 46k

that is 50/50

Yes but the thing that makes it tough to understand is the fact that the contestant can only choose 3 of the 4 suit cases, and he will never know when which ¤-amount is where.

Please try explaining it to someone who really does'nt see the "correct" point of view and really believes he is right.

omg :D

Lol. If he can't understand it now, he never will...

Doesn't matter that he only can choose between 3, all 4 suitcases are in play and we dont know what is in any of them.

This is dumb and totally basic. I don't see why you want to "win" the argument if they can't understand what was written above... They clearly have no idea of basic math/probability... but let me explain so that I think that even they can understand:

There's 4 suitcases left; 2 have over 45k and 2 under. This makes it a 50/50 proposition. Sure, u can argue: But you can only choose from 3, what if the suitcase with 100k is my own (ie I can't pick it now). If we _knew_ that we had the 100k suitcase it would chance the probability to 1/3 for picking a suitcase with >45k and 2/3 for picking one with less than 45k... but the truth is, that we DONT know anything about our own suitcase, hence it's equally likely that our own suitcase contains an amount higher than 45k that it is that the amount is lower.

If you want it calculated we can do it (this is redundant, the answer 50% is trivial) from your point of view:

P(amount picked > 45k)
= P(own suitcase = 100k)*P(amount picked > 45k given that own suitcase = 100k) + P(own suitcase = 50k)*P(amount picked > 45k | own suitcase = 50k) + P(own suitcase = 10)*P(amount picked > 45k | own suitcase = 10) + P(own suitcase = 0.01)*P(amount picked > 45k | own suitcase = 0.01)
= (1/4)(1/3) + (1/4)(1/3) + (1/4)(2/3) + (1/4)(2/3)
= (1/4)(6/3) = 1/2 = 50%

(the "|" means "given that...")

Here I have taken into account that our suitcase can contain different amounts... it follows that it doesn't matter since we don't know anything about our suitcase (we are equally likely to have picked either of the 4 numbers for our own suitcase - 25% or 1/4).

Very good explanations so far! I do not think it is possible to make it more comprehensible then Bender's answer!

What color are these suitcases, and can they be opened without a key?

Hey...

50% is of course the right answer.
If I have 2 apples and 2 bananas in 4 bags, the chance of me picking a banana is 50%, even if i carry one of the bags on my back.

BTW, If u finally manage to convince ur buds about the 50%, consider making a bet concerning "Schrödingers Cat" being both alive and dead at the same time

All of this assumes that the bank is unaware of the amount in the suitcase you are holding, or does not take it into consideration. As it is usually in the best interest of the show to keep going as long as possible, the bank offering more than the average value of the suitcase (and thus tempting the player to stop) might suggest otherwise.

Should their offer depend on the amount in the player's suitcase, the probability for opening a small amount might be greater than 50%. Of course, they could also be bluffing...