Deal or No Deal Mathematical Probabilities

Factual Questions
1

My wife and I are fans of the show Deal or No Deal (yeah, live with it).

I’m interested in strategy in the game and the probabilities involved.

For those unfamiliar:

The game features 26 suitcases, each containing one of the following monetary values:

$.01
1
5
10
25
50
75
100
200
300
400
500
750
1,000
5,000
10,000
25,000
50,000
75,000
100,000
200,000
300,000
400,000
500,000
750,000
1,000,000

I pick a case to hold onto, unopened, and then open up the other cases one by one, revealing which prize I didn’t win. At certain points, the “banker” who is paying out whatever amount I finally end up with makes me an offer to buy me out of the game.

The amounts are randomized among the cases, and neither I nor the banker knows what’s in each case at the beginning.

Since they are random, I will not confuse the issue by randomly choosing numbered cases. I choose case #1 to hold onto.

Round 1: I must open 6 cases, so I’ll just go in numerical order.

#2 - 75,000
3 - 100
4 - 400,000
5 - 400
6 - 500
7 - 5,000

So I don’t have any of those amounts in my case. The banker now makes me an offer to buy my case and walk away from whatever amount might be in there.

The offer is $28,064.

The remaining possible amounts left to be revealed are:

$.01
1
5
10
25
50
75
200
300
750
1,000
10,000
25,000
50,000
100,000
200,000
300,000
500,000
750,000
1,000,000

I must open five more cases before another offer is pending.

My question: what is the probability that I will:

a) beat the banker’s offer?
b) have a dollar value in my case greater than or equal to $100,000?
c) end up holding the $1,000,000 case?

Is it worth it to me to open 5 more cases before hearing another offer, knowing that if I eliminate larger amounts from the board, the offer will drop, but if I eliminate smaller amounts, the offer will go up.

I suspect it is something like the fabled Monty Hall problem, but in that one, Monty knows what’s behind each door. Deal or No Deal’s banker does not.

Well?

2

I should add that the order of revealed values and banker offers are straight from the online version of the game, which differs from the TV offers, AFAICT, only in that they are not rounded off to the nearest $50 value.

3

We went through this so you might try searching the board if you haven’t already. Long story short, the offer the banker gives you is pretty much the expected out come. That is, if you run the simulation a million times you will average what ever the banker offers. The game, and strategy if you want to call it that, is a simple matter of risk tolerance. That is, how much guaranteed money is the possibility of getting more money worth? In that way, there is no true way to beat the game or even play it more skillfully.

4

For a start, you have a 1/20 chance of $1,000,000 (which would be worth $50,000 alone). Unless you value the certainty of $28,064 more highly.

But I don’t understand how you decide when to stop. After you have rejected the offer of $28K, can you just say that you are sticking with whatever is in the present case? Or do you have to open more?

5

Didn’t find a previous thread on this but I’ll try diferent search terms.

6

The only way to stop the game is to accept an offer or run out of cases to open. If I reject the 28 grand, I must open 5 more cases, then weigh another offer.

7

Sorry, meant to address this as well. Is it really 1/20?

In the Let’s Make a Deal scenario (and if you want to discuss that again, please go to one of the many recent threads on the subject, don’t discuss it here), the probability of you having picked the prize door initially never changed from 1:3, even when Monty opened a non-prize door for you.

I presumed, based on that, that my probability of having picked the $1,000,000 case never changed from 1:26. If I’m right, how does the remaining 25:26 chance that I don’t have the million distribute itself among the remaining cases as they get eliminated? But maybe I’m wrong

Of course, on the show, no one has walked away with the million, or even half that yet, that I’m aware of.

The banker’s offer is based on some computer-generated evaluation of risk tolerance, i.e., what’s the smallest amount of money we can pay out in order not to risk having to pay out the largest amount possible?

Since those numbers are available to them, and neither of us has the advantage of knowing what’s in each case until it’s opened, they must also be available to me. The only info they have that I don’t is the precise formula by which they calculate their offer.

What is the actual mathematical likelihood at this point in this particular game that, by eliminating five more values at random from the list, I can get a better offer from the bank, knowing only that if the majority of the eliminated values are toward the low end of the list, the offer will tend to go up, and if a majority of them are on the high end, the offer will tend to go down?

8

I dislike math, but it sounds like it is just probability.

My question: what is the probability that I will:

a) beat the banker’s offer?
b) have a dollar value in my case greater than or equal to $100,000?
c) end up holding the $1,000,000 case?

a) I’d guess you would take the total value of the 20 cases, add them up and divide by 20 to get the banker’s value. seven are above the banker’s value and 13 are below it. However the values above the banker’s value are far higher (500k vs 28k) as opposed to the values below it ($5 vs 28k). So on that one turn you have a higher probability of finding the case below 28k, but the cases above 28k can do more to increase your total value.

b) add the cases together, there are 20. Six have 100k or more in them. So the probabiliy is 6/20 = 30%

c) 1/20 = 5%

9

Sounds good to me. I can’t seem to get search to work today, but I did find this page with a dowloadable spreadsheet calculating expected value with each opened case.

His spreadsheet says my expected value at this point is $146, 871, so I press on.

In Round 2 I pick 5 cases:

8 - 50
9 - 25,000
10 - 100,000
11 - 50,000
12 - 300,000

The offer is now: $36,393.

The remainin values are:

$.01
1
5
10
25
75
200
300
750
1,000
10,000
200,000
500,000
750,000
1,000,000

Spreadsheet boy says my expected value is: $164,158

Is he right? do I press on? I have to open 4 cases before another offer is extended.

(And in the Monty Hall Problem, the 2:3 chance my first pick was not the prize essentially consolidated on the last door. How does that NOT apply here?)

10

The spreadsheet is exactly right, you press on. This assumes, of course, that you are either planning on playing the game hundreds of times, or you are rich enough to be risk-neutral. However, this completely ignores the real rules of the game.

In the real game, you can only play once. What’s more, you have to send in a video audition to get on the show, as well as certify that you are not a professional gambler. Presumably, only poor (or at least middle class) people are chosen to be on the show (and this seems to be the case in the episodes I have seen). This makes the decision much more interesting. 60 or 100 grand would make a huge difference in the lives of the people who are on the show, and thus are not amounts they can afford to turn down lightly. Of course, everyone should turn down the first few offers, as they are always ridiculously low. Things don’t get interesting until around the fourth offer.

Monty Hall knows which door has the prize, and never picks that door. This makes all the difference. If Monty were choosing randomly, then you would have something more similar to the deal or no deal situation.

This is completely wrong. The banker’s offer starts out far lower than the expected value and ramps up towards it only slowly. Near the end, it may even exceed the expected value. This means there is some additional strategy in deciding to wait for his offer to reach the expected value or not. Now, I agree that this isn’t much strategy. It’s so simple in fact that I can’t stand watching the show (I do it as a favor to my girlfriend). But at least calculating the expected value in my head and comparing it to the current offer gives me something to do to distract myself.

By the way, did anyone watch Bobby’s world with Howie Mandel as a kid? I loved that show! Actually that’s more of a cafe society thread…

11

Spreadsheet boy is right, in that your expected value is indeed $164,158.
Whether you should take an offer less than that is another question, not completely simple to answer.

If you could play this same game hundreds of times and get in the end the average win, then you should press on at this point, no question, the highest average is what you’re shooting for.

But you can’t; you only get one shot. In this case, there are some personal values that come into play. First is how much more than first $30,000 of winnings means to you than the last $30,000 of $1,000,000.
In the extreme case, if Vinnie is going to shoot you in the head tomorrow if he doesn’t get $25,000, and your uncle Bill Gates has just secretly named you his sole heir (and confided about his incurable and quickly-progressing illness), then that first $25,000 means a whole lot, but winning $975,000 more is no big deal. In that case, the right thing to do is always take the bank offer.
Even absent extreme circumstances, for most people it’s worth a little bit of expected value in order to reduce risk (that’s what home insurance is all about). The question is, how much (in other words, how comfortable are you with the chance of getting nothing?).
The difference between expected value and the bank offer is pretty large here, so I’d probably go for it, myself.

12

I think if you actually computed it, the early offers are always significantly less than the expected value of the cases and you should always reject those. The offers get closer to the expected value of the cases as the number of cases get smaller, finally becoming equal to the expected value when there are 2 cases and you have to decide to take the offer or the case you chose.

You may want to take one of the offers for less than the expected value though to eliminate the risk that you would win essentially nothing. For instance, consider the lady that had $300,000, $5, and $1 left and was offered $80,000. This offer was $20,000 less than the expected value, but it could be argued that you should take it unless $80,000 means nothing to you. She rejected the offer, opened up the $300,000 case and wound up winning only $5.
The Monty Hall “paradox” does not come into play here at all because no one knows where the prize is. Suppose you picked case 10. You then opened 6 cases and none of them were the million dollars. If another person now could pick any case (including yours), the chance that they would pick the million dollar case would be 1 in 20. Since you don’t know any more than they do your chance of having the million dollar case is exactly the same (1 in 20).

On preview I see since I was interrupted while writing this that a lot of what I said has already been said.

13

But you elaborated on Emerald Hawk’s point about how the probabilities are essentially “reset” each round, which helps a lot, so I thank you.

14

So now it’s round three.

I must pick four cases.

13 - 200
14 - 300
15 - 200,000
16 - 25

My exected value is now: $226,109.

The offer is: $62,964. If I turn it down I must open three cases before the next offer.

The remaining possible values are:

$.01
1
5
10
75
750
1,000
10,000
500,000
750,000
1,000,000

If my memory of probability serves me correctly, I calulate about a 34% (8/11 * 7/10 * 6/9) chance of only removing small offers in my next three case picks, whereas the likelihood of knocking out all top three values in the next three picks is about 0.6% (3/11 * 2/10 * 1/9). Correct? If so, I press on.

15

Actually, that’s not true, is it? There’s one case that has no chance of being opened, mine.

So if I don’t hold the million dollars, my chances of knocking out the top three values are 3/10 * 2/9 * 1/8 = 0.8%

If I do hold the million dollars, what are my chances of lowering my offer by knocking out the other two with any two of my three picks?

And given the likelihood of my holding the million, what is my overall likelihood of getting a lower offer? My memory of the math escapes me.

16

The game is actually a very simple premise, expanded to seem complicated and confusing, and has the human risk/reward factor going for it.

For example, if I gave you the choice of picking one of two boxes, a $5 one and a $5,000 or taking $2,500 and walking, what would you do?
Depends on the individual and how much of a gambler they are. Is loosing $2,500 worth the risk of a 50/50 chance of doubling it? Depends on how often you come by $2,500.

How about $500 and $50,000, walking away with $25,000? Again it’s a personal choice with very simple odds. Not a whole lot of math there to confuse you. No real answer as to a smart or dumb decision. No real strategy. Just a choice to gamble or not.

How about $5,000 and $500,000, walking away with $250,000. Again, nothing too complex here. What’s a foolish risk and what’s not? Same odds as previously, just bigger amounts that you don’t come by everyday.

So, they just took this simple personal gambling question and made it really complex and confusing as to appear to have a strategy and smart/dumb decisions when in reality it’s just a lot of personal choices as to how much money are you willing to give up for a chance for more money.

17

At the start you pick and hold one case containing one of twenty amounts ranging from $0.01 to $1M. You now engage in playing the game by opening one of the remaining 25 cases with a 1:25 chance (probabibity) that it contats the BIG one or 24:25 it doesn’t. After opening 6 cases “The offer is $28,064.”
You have only two (2) options: “DEAL”

18

:smack: Typing one handed with a broken wrist in a cast & hit somethig… :smack:

OR “NO DEAL” and five cases to open.
When you reject the Banker’s offer then what?
Open more cases to continue OR do what no one has done yet. Walkaway and go home!