Probabilities In "Deal Or No Deal" ????

Question

Help me understand this:

Someone wrote that the probability of winning the top prize of one million dollars in "Deal Or No Deal" is 1/56.

I don't see why this is so.

I think you either win or fail to win the million dollars right at the beginning of the game.

And, at the beginning of the game, there are 26 cases. So, the probability that you will win 1 million is 1/26.

So, if you decide to play the game to the very end and say NO DEAL to every offer, then you win 1 million dollars. There is no other way to win it.

The fact the you chose 1 million dollars in the beginning is independent of what happens after that. Opening the other cases does not put you at risk of losing the case that you chose. Your case doesn't change.

So, how is the probability of winning 1 million not 1/26?

Remember, the banker will never offer you 1 million. If you have 2 cases left and one is the case with 1 million, the banker may offer you something like 600,000.

Comments??

Answer 1

There's only one event where you can win a million.

You select the case (1/26) at the beginning and then successively open each of the remaining cases, revealing that you had the million all along.

In the long run, 1 out of 26 people will have the winning case, but after let's say 793,857 contestants (by then Howie will be wishing he had hair to pull out), even less will end up opening up 25 cases to get to the million they didn't know they had.

And that's great. Because over time, the average expected winnings are only $131,478. Playing online, I have only earned $102,000 playing conservatively (the-bank-offer to remaining-expected- winnings ratio not exceeding 90%).


Play the online game, and analyze the offers on a spreadsheet as you go. You'll find the ratio starts at about 20% and slowly climbs to about 50% for 6 rounds and then jumps up to about 90% in about 2 rounds.

Know when to quit.

Answer 2

This is a joint probability thing.

If I understand the game, the million must be in the first choice, but one has to go through 26 choices before going back to find out if the million was in that first choice.

Thus we have P(s = $1million) = n(s)/N(m) = 1/26; where n(s) = the number of ways we can succeed on each choice and N(m) = the number of all possible outcomes (including s <> $1 million) on the m choice = 1, 2, 3,..., 26.

Thus, to win, we must choose the right case (s = $1 million) on the first choice and NOT choose the right case (f = s <> $1 million) on the remaining 25 choices. The probability of not choosing the right case is P(f = s<>$1 million) = n(f)/N(m); where n(f) = the number of ways to fail choosing the $1 million case.

The probability of choosing s = $1 million on m = 1 is P(s = $1 million) = n(s)/N(1) = 1/26, as some have indicated.
The probability of NOT choosing s on m = 2 is 1 - n(s)/N(2) = 1 - 1/25 = 24/25; for m = 3 it's 23/24; for m = 4, 22/23, etc. until m = 25 which is 1/2.

Thus the probability of winning the whole kaboodle is P = (1/26)(24/25)(23/24)(22/23) on down through (3/4)(2/3)(1/2) = (1/26)(1/25), which is simply the probability of succeeding (s = $1 million) on the first case times the probabilities of not succeeding on subsequent cases. So you have a 1/650 chance of winning the big bucks...that's actually good odds, much better than roullete, horse racing, etc. [See source.]

Answer 3

I do not think that is totally correct. Because, once you choose the case to be saved until the end of the game, you move on to the next level of the game. Choosing cases, one by one, to be cancelled out and if you go through all of them and none is the million then it has to be in the very first case you chose and you win it. So the odds of you selecting that million is 1/26 but after that the odds of that same million being up on the board are 1/25 for the next choice. Then 1/24 for the one after that, and it will continue until it appears or there are no cases left.

Answer 4

The probability of selecting the $1M case is 1/26. However, Deal or No Deal is a several move game, and the actual probability of continuing to eliminate cases depends on the risk preference of the player (risk loving, risk neutral, or risk averse) and the method for calculating the bank offer. Standard probability modeling assumes risk neutrality. That means for example, if $0.01 and $1M were the last 2 cased on the board and the bank offer was $500k, the risk neutral contestant would be exactly indifferent. Assuming the bank simply offers the probability weighted average of the unopened case values, the risk neutral contestant would stall and need to flip a coin. Risk loving contestants would go for it, and risk averse contestants will take the bank offer. In real life, contestants risk preferences change with the perceived value of the bank offer and the excitement of the game. For example, someone who makes $100k/yr but is generally risk averse may deny a $25k bank offer, but wouldn't tempt fate above $100k.

Answer 5

You are correct, the chances are 1 in 26 not 1 in 56.

Answer 6

The probability is based on not "knowing" what is in the case you chose. Basically it is the probability that you have chosen the case with 1 million dollars. Until you look and see what is actually in the case it is just a probability.

Answer 7

I have never seen the show, but from how you described it, your logic makes sense.

Unless they're including the odds of being selected from the audience.

Answer 8

I totally agree with you.
Now, someone should determine the odds of actually getting on the show.

Answer 9

the banker gives only the odds and the% of the canes of haveing the right sucast nad he wan,t tobuy the seucate and cheapply as posilble.

Answer 10

just remember that it is a game of chance. lucky for you if you get the grand price