Well, I wasn't offering a strategy for playing the game. Part of the discussion that took place on this thread involved the "Monty Hall" game, and whether of not DOND was an example of such a game (it isn't). My comments are just to help people who might stumble across the thread get terminology correct.

As far as playing the real game, I agree with you.

It would be helpful if we knew the banker's formula, but we don't. Here on this page I have two version of the formula used by the on-line game, but neither matches the real game.

http://www.davegentile.com/stuff/Deal_or_no_deal.html

Dave

The game is pretty simple, you play for the offer. 99% of the time you don't go down to the wire and choose your case and the last case. 99% of the time you take an offer.

Odds and probability are over-rated in analyzing this. As long as you have some big numbers on the board, you're going to get a big offer.

The longer you go, the better the offers get.

The key is simple, keep playing while several large numbers are on the board, quit when there are few. Don't keep playing if there is a chance that one of the big numbers will be eliminated.

Bayes and all that other garbage won't help because the offers are not based strictly on probability. And you always take an offer.

Yes we pretty much resolved the whole thing a while back, but every once in a while the thread comes back to life. I've followed it for awhile now.

My background: I have a couple of quantitative degrees, and I currently work as a statistician. I'm (still) working on publishing a paper related to probability theory, and (just by coincidence) I happen to be currently taking a probability theory class right now and living and breathing the stuff (with the goal of adding another degree or and least a certificate to the list).

O.K., so here is how I would describe things.

According to classical probability and statistics, the probability of a case containing the 1,000,000 is 1 in 26, and that never changes. Your pick is a random variable, and does not change.

However, in classical probability and statistics you can then talk about a "conditional probability". Once you have opened 24 cases and not found the 1,000,000 the conditional probability of winning 1,000,000 GIVEN the 24 open cases is 1/2.

There is also another school of probability thought called "Bayesian" probability. Here the probability at the end would be 1/2.

Your distinction about "odds" vs. "probability" is not correct, however. Odds are just a simple calculation you can make from the probability, which just states the same thing in a different way. A horse might have a 1/6 probability of winning and we say the odds are 5:1. This is just (1-p)/p.

However, with a little more detail, your distinction could be made correct. "Odds" are most often associated with gambling, and so is Bayesian probability. It is well established in decision theory.

So if you say the "classical probability" is 1/26, and the "Bayesian odds" are 1/2, you would be correct.

But the clearest statement would be to say that at the end the conditional probabilty is 1/2. Everyone in both schools would be clear as to what you meant.

Just trying to keep the thread informational at this point.
)

Dave

To answer the brother vs. engineer question correctly, you need two basic things:
1. An understanding of the rules of the game (yes)
2. An understanding that there is a difference between probability and odds.

So here we go:

You choose a case from the board to be yours. The PROBABILITY that this specific choice yielded a case with $1M in it is 1/26. This will NEVER change.

Your current ODDS of your case containing the $1M are 1:25

The PROBABILITY of you winning $1M based on the next case you open is 0 and does not improve because the most you can win at this point in the game is what the Banker offers.

Each successive case that you remove that doesn't contain the $1M improves the ODDS of your case containing the $1M, but not the PROBABILITY because the PROBABILITY that your case has $1M in it is linked to the single event of you choosing that case.

If you make it all the way down to the point in the game in which there are only two cases remaining then AND you refuse the bankers final offer:
1. PROBABILITY of your chosen case having the $1M is 1/26
Because this choice happen only once
2. ODDS of your case having $1M is 1:1
$1M vs. not $1M
3. PROBABILITY of you winning $1M is 1/2 (Here is the 50/50 chance that everyone gets hung up on)
take your case, trade for the othercase
4. ODDS of you winning $1M is 1:1
$1M vs. not $1M

You can't actually calculate the probability of winning this game because you can't accurately calculate what each individual who plays will consider a good deal and hence you can't calculate the probability that they will take the deal at any give point

That's the thing. The offers are not strictly based on averages. They obviously are to some extent. But if you figure that after the first 6 boxes are removed and the $1 million and $500,000 are still there that:

$1.5 million/30 = $50,0000 alone

So that means as long as $1M and $0.5M are always on the board, an average would never be less than $50,000.

But you know that's not the case. Initial offers are usually like $20,000 and most times LESS.

Offers reward you for moving ahead. And typically if there's a large offer, that means you have a good shot of increasing the offer with another box.

You have to be willing to have a little wiggle room. Let's say you get a $75,000 offer, but you know the chances are high of eliminating another small box. But if you choose a big box, the offer may go to $50,000. Why not take the shot?

I compare it to jumping on lily pads in a pond. Do you move ahead or stay safe? How safe is the next box? What are the rewards/losses for jumping to the next lily pad. If it's relatively safe, i.e. your offer won't drastically change, then do it. Otherwise stay put.

Actually, I just scrolled up and saw that I did some math before that does apply to this-

11 cases remaining: (under 30k average) 3.2%, (30k-80k average) 18.6%, (80-180k average) 57.5%, (over 180k) 20.7%.

Going by those approximate numbers, about 78% of the time you'll have an offer of 50k or higher with 11 cases remaining (this assumes that an 80k average will get you a 50k offer).

So, you have a 78% chance to succeed at just that one point in the game alone. In the other 22% of the games, you'll have many other chances both earlier and later in the game to reach the 50k mark.

Maybe my guess of 95% was high, but I would be surprised if you don't hit 50k in over 88% of the games.

I will agree with one thing... if you eliminate 5 very low cases to start and immediately get an offer of $50k, it's much worse to choose to stop there. This is because early offers are a very low percentage of the remaining case value on the board (sometimes like 30% of the average), when later on the offer will be 70-80% of the average or even more.

This is the easiest strategy, IMHO, and it seems to work 99% of the time:

As long as you have at least 2 boxes above $50,000, then keep guessing boxes. Once you get to 1 box at a time you can either keep going until only 1 box above $50,000 is left OR take an offer that seems like a good deal.

The only exception to this is if you are still choosing multiple boxes (4-6 at a time) and you've almost wiped out the entire right side. Then I would jump on a big offer.

The game is really not that challenging in terms of how to play. When I play online, I don't even look at the boxes it picks. I just randomly pick boxes until it gets down to 1 box at a time. Then I figure out when to take the deal.

The goal is to maximise the chance of YOU winning. (Minimize risk)

To play the odds alone is to maximize the chance of the HOUSE losing. (Maximize payout)

One thing I can say absolutely is I would never pick another case if there were just one high value case left. When being the unlucky person (even if it's 1/10) is not worth the risk, even if it's a crappy offer. Theres nothing crappy about free money.

I think you vastly over estimate the ability to see $50,000 as an offer. And the way you talk, you sound like you're just making up these statistics based on what you ASSUME and not what you KNOW.

Realize that the bank offers are not predicated on an average anyway otherwise you'd see $100,000 offers at the very beginning. And you get rewarded for continuing to play.

So if you get a $50,000 offer early, that means more likely than not, you have a REAL probability of winning more if you continue. So why screw yourself and quit early?

The average winning when you start is around $100,000. So if everyone played until the end, on average the bank would lose $100,000 per person. YET, initial offers are usually miserable, even when you eliminate many smaller amounts.

So if you realize that your possible winnings average has shot up to say $200,000. Why would you stop?

The bank offers are not averages. Early offers are less than the probabilities of what you could win. So stopping early is a mistake if you are getting large offers like $50,000.

Likewise, if you have 5 boxes under $1000 and 1 box at $100,000, taking an offer of $20,000 is a smart move. Because that is better than your real odds.

So sticking to a preset number is the wrong strategy. Because if you get that offer early, you're hurting your likeliness of winning more. And if you wait too long for that offer, you're increasing your chances of walking away with less than $1000.

You need to play the board and know your odds. Quit earlier when the board is against you, keep going when it's in your favor. But it seems to make sense that you should keep going to at least the rounds of guessing 1 box at a time. The only time that wouldn't make sense is if you eliminated most of the high value boxes.

About the "pay $5000 to play", it is virtually guaranteed that at some point (most likely your first offer), you will see an offer of over $5000. Just take that and quit the game.

About the "I'll quit when I see a $50k offer", you're calculating it wrong. Just because only 9/36 of the cases are worth 50k or more, that does not mean there is a 9/36 chance of seeing an offer of 50k or more at some point during the game. In fact, the vast majority of games have an offer of 50k or higher at some point.

"you shouldn't dump a strong position in the game (all large values on the board) because $50,000 is a lot of money to you"

This would never get to happen. To reach the position of having all large values on the board, you need to turn down offers when the board is 2/3 large 1/3 small, for example. During this time you would have received an offer of over 50k and taken the deal, so you would never get a chance to see the "large cases only" board.

"1) you get bad luck "

This is possible. But the player has chosen to accept the approx. 5% chance of failure in exchange for the 95% chance of getting at least 50k.

"2) you jump on the first offer over or near $50,000, this is a bad move if you're in a great position on the board"

Again, if your offer is in the neighborhood of 50k, you will never be in a great position on the board. First because playing the 50k strategy means you never reach a "large cases only" situation, and second because if you were in that situation the offer would be much larger than 50k.

"Based on your suggestion, someone may have $25,000 credit card debt and jump on the very first offer."

Well, it all depends on their situation. If this debt has been ruining his life, he might take the offer. If he makes $80k per year and has $50k in investments he could sell and pay off the debt anytime he wanted to, he's much more willing to take risks in order to get more money. And if he's Bill Gates, the optimal strategy is to skip the entire game and just keep your selected case.