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Monty Hall paradox on Deal or No Deal

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If you're a contestant on Deal or No Deal, and you're down to two cases, you're given the option to switch your cases.

 

I know that, mathematically, it is a good idea to switch the cases, due to the Monty Hall paradox; however, would your chances of getting the better case go to 1/13, or 25/26?

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The odds go up in the Monty Hall problem because the host knows which door has the better prize and eliminates a different option purposefully. In Deal or no Deal the host doesn't know which case has the larger amount. Even if he did, the host doesn't eliminate a choice on that game show. There is no advantage to switching when 2 cases are left. Both cases have a .5 probability of being the larger amount before and after switching.

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But in Deal or no deal, the host does know which box has the smaller prize, albeit at the same time as the contestant.

The host or 'Banker' doesn't always offer the option to switch unless the third-to-last box is a low number.

 

e.g.

There are three boxes left [1p] [£1000] & [250000]

If the contestant opens the [1p] box the banker will offer the contestant the chance to switch, if they open either of the other two, he wont offer the switch.

Would the Monty hall effect still work if the host knows which one of the boxes which has the lower prize?

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But in Deal or no deal, the host does know which box has the smaller prize, albeit at the same time as the contestant.

The host or 'Banker' doesn't always offer the option to switch unless the third-to-last box is a low number.

 

e.g.

There are three boxes left [1p] [£1000] & [250000]

If the contestant opens the [1p] box the banker will offer the contestant the chance to switch, if they open either of the other two, he wont offer the switch.

 

I didn't know that, but I don't watch the show. It seems odd--only if the third to last case is the lowest amount :confused: Hum.

 

Would the Monty hall effect still work if the host knows which one of the boxes which has the lower prize?

 

No. Neither the banker or host are choosing cases to open or making any move based on secret knowledge. Even if the banker only offers a switch if the 3rd to last case was the small amount--the player has that info as well, so it doesn't reveal anything. It doesn't help constrain the probability of the last two cases.

 

Monty Hall would only work if the host decided which case to open knowing that it could not be the larger amount--effectively narrowing down the possible location of the largest amount to 2 out of 3 cases rather than 3 out of 3.

 

Edited To Add--

 

Here is a comparison I found on wikibooks:

 

When only three cases remain, Deal or No Deal might seem like a version of the Monty Hall problem. Consider a Deal or No Deal game with three cases (similar to the three doors in the Monty Hall problem). The contestant has one case. Then, one of the two other cases is opened. Finally, the contestant is given the option to trade his or her case for the one unopened case remaining.

 

 

The Monty Hall problem gives the contestant a 2/3 chance of winning with a switch and a 1/3 chance of winning by keeping his or her case. However, there is a critical difference between Let's Make a Deal and Deal or No Deal. In the Monty Hall problem, the host has used his secret knowledge of what lies behind each of the three doors to cause a bad choice to always be revealed. This non-random selection of a bad choice is what causes the difference in odds of winning between switching and not switching on Let's Make a Deal.

 

 

In Deal or No Deal the odds behave exactly as you would instinctively expect them to: 2 boxes with a fifty-fifty chance of the top prize being in either of them.

Introduction to Game Theory/Deal Or No Deal

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I see.

but, if when the third-to-last case was opened and when that case had the lowest amount.

if we (As the audience) were to observe the banker consistently offering the switch. might that indicate that the banker has foreknowledge of which box contains which amount?

That is, in this scenario, the contestant is effectively opening the low-amount-box on behalf of the banker.

The banker knows what is in every box and so invokes the Monty Hall.

 

Also, is knowing where the prize isn't as advantageous as knowing where the prize is?

 

I can't see the real difference between 'make a deal' and 'deal or no deal'

In both cases

1 contestant chooses a box

2 bad box is opened

3 host offers a switch

Edited by tomgwyther

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That is, in this scenario, the contestant is effectively opening the low-amount-box on behalf of the banker.

 

The choice of which case to pick belongs to the player. When the player has three cases remaining they choose one without knowing which of the three amounts is in the case. There is an equal (1/3) probability that the case contains each of the three remaining denominations.

 

The case is opened and the amount revealed and everyone now knows which of the three amounts it contains: the lowest amount, the middle amount, or the highest amount.

 

Apparently the banker then offers the player the option to switch the remaining two cases if and only if the previous case contained the lowest amount. Everyone, including the player, knows if the previously opened case held the lowest amount so the banker reveals nothing.

 

In Deal or no Deal the act of switching does not increase the odds of opening a better valued case. In Monty Hall, switching gives better odds.

 

The banker knows what is in every box and so invokes the Monty Hall.

 

It is not enough for the banker to know what's in a case. It might be most helpful if I just describe the Monty Hall problem and it should be clear how Deal or no Deal is different.

 

The Monty Hall problem has a contestant choose one of three doors. Behind one of the doors is a car. Behind each of the other two doors is a goat. The player is trying to choose the door which hides a car and they make a choice 1, 2, or 3. The chosen door is not immediately opened. The host instead opens a different door which he chooses himself (out of the remaining two) revealing one of the two goats.

 

The host knows where the car is and that knowledge allows him to choose a door with a goat. There are now two closed doors one of which the player has already chosen and an open door which revealed a goat. The host then asks the player if they want to switch their initial choice and instead pick the other unopened door.

 

In this situation it is advantageous for the player to switch. Going with the original choice will have a one in three chance of getting the prize. Switching results in a two in three chance. Most people intuitively think that switching will make no difference. But, think of it like this:

 

When the player first picks a door each of these situations is as likely as the other two:

  1. The chosen door holds the car
  2. The chosen door holds goat A
  3. The chosen door holds goat B

If (1) happens then the host will reveal either goat A or goat B. Switching means the player loses.

 

If (2) happens then the host must reveal goat B since he never reveals the car. Switching then results in a win for the player.

 

If (3) happens then the host must reveal goat A. A switch then results in a win.

 

Without switching only one of situations 1, 2, and 3 get the car. With switching two of the situations get the car. Switching therefore increases the chance of success from 1/3 to 2/3.

 

Also, is knowing where the prize isn't as advantageous as knowing where the prize is?

 

Knowing where the prize is means the probability of winning is 1. The choice is always successful no matter how many unopened boxes there are. Knowing that a box is not the prize is only that advantageous if there are only two options.

 

I can't see the real difference between 'make a deal' and 'deal or no deal'

In both cases

1 contestant chooses a box

2 bad box is opened

3 host offers a switch

 

In Monty, the box in (1) is not revealed. In Deal, (2) is not always the case. In Monty, (3) is an advantageous move. The wikipedia page: http://en.wikipedia.org/wiki/Monty_Hall_problem might explain better than I have.

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T The wikipedia page: http://en.wikipedia.org/wiki/Monty_Hall_problem might explain better than I have.

 

Actually, you've explained it better than the wiki page (Which I read top to bottom previously)

I was unaware of some of the finer points of the Monty Hall game, namely at what point in time choices occur, and when boxes are opened/chosen and by whom.

Thanks for the explanation, I love to learn. To be proven wrong as as insightful as to be proven right.

 

I wonder, are there any other mathematical probability anomalies which would make a good TV game show?

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Actually, you've explained it better than the wiki page (Which I read top to bottom previously)

I was unaware of some of the finer points of the Monty Hall game, namely at what point in time choices occur, and when boxes are opened/chosen and by whom.

Thanks for the explanation, I love to learn. To be proven wrong as as insightful as to be proven right.

 

Thank you for the compliment. I admire your approach, and agree--you can't go wrong when you're willing to be proven so :)

 

I wonder, are there any other mathematical probability anomalies which would make a good TV game show?

 

Not sure about game shows, but certainly my favorite game of mathematical probabilities is poker :D

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