Does the Monty Hall Paradox Apply to Deal or No Deal?
September 1, 2009 by Admin
Filed under Simulations
The host looks Jack in the eye.
“Jack, think about your situation for a moment. A million dollars is a lot of money. It’ll change your life forever. There are only two cases left. One contains a single dollar and the other a life changing amount.”
“Jack you’ve gone all the way, you have turned down every offer.” The host pauses dramatically, “Do you want to switch cases?”
Jack feels the sweat trickle down his brow. For a second he remembers his statistics class and that whole problem with Monty Hall. but did it apply to Deal or No Deal?
The Debate
After several discussions with other friends, I discovered that there was still much debate as to whether or not the Monty Hall Paradox applied to ‘Deal or No Deal’. Lot’s of it heated. As heated as Jon and Kate separation! So to settle it I just had to find out myself with empirical evidence.
Let’s Define Some Terms
MHP: Monty Hall Problem/Paradox. If you don’t know what it is check out this YouTube Video for a quick explanation. If you already know it, skip the video.
MH Camp: Those who believe MHP does not apply to DoND.
DoND: The game show Deal or No Deal.
DoND Camp: Those who believe MHP applies to DoND.
RE: Random Elimination, the process in which cases/doors are eliminated randomly as in DoND
FE: Forced Elimination, the process in which non-winning cases/doors are eliminated simply because they are a non winning door.
It Does Not Apply Camp
In the MH Camp, the main argument is since the host knows where the prize is and does not pick it gives the statistical advantage. In DoND the advantage does not exist becase choices are eliminated randomly.
It Does Apply Camp
The main argument from the DoND Camp is it doesn’t really matter whether the host knows where the prize is. Assuming that you’re in a scenario where you’ve successfully eliminated all other cases leaving the million in play. The result should be the same as MHP whether or not choices are eliminated through FE or RE you should swap.
Our Simulation
The real difference between Monty Hall and DoND is that MH has a Forced Elimination (FE) of a non-winning door. DoND there is a Random Elimination (RE) of a case/door. Basically we simulated playing out thousands of games using FE and RE and recording the results.
The goal of the simulation is to test whether or not it Monty Hall can apply to Deal or No Deal during times where the million is in play and the contestant is asked to switch cases. If it’s true that in situations where the prize is still in play when you get down to the final two cases the results would be similar to the MHP. It should be greater than 50-50 to switch.
Our First Test Run
We played 120,000 games. 26 cases, Deal or No Deal rules. We looked at all the games that had two cases left and the million remaining in play. We look at the probability to switch. The results were… 50-50 to switch.
Combing through the data and looking for any kind of error none. The results were right and accurate. It is 50/50 to switch.
So what accounted for the difference? After analyzing the data I felt the simplest and most effective way to display our results in an easily understood manner is to modify the simulation, slightly.
A Second and Third Simulation
I decided to run two simulations using only three doors. First, I ran 10,000 games using DoND Rules.
Deal or No Deal Rules All Combinations - the winning case (case #1, the numbers #2 and #3 are non-winning cases) is picked 33% of the time as expected when you have only 3 cases to pick from
Like MH the winning case was chosen 33.25% of the time. Now if we only count the games in which the winning case remained in play. This is the result.
Notice the change in the percentages when you only count games in which the winning case remained in play
Notice that the percentages change to about 50-50 (or 49.90% and 50.10%) to switch. Since we are only counting the winning positions the number of games drops to 6,663.
Then I played out 6,663 games (to match the DoND number of games where the prize is still in play). Here is the breakdown of all possible result combinations and the count each occurrence. The results are as follows.
Monty Hall Forced Elimination Results
As you can see certain combinations occur more often in the Monty Hall Rule Simulation versus the DoND Rule Simulation. Every combination in the DoND Simulation is spread evenly. Whereas, in the MH Simulation the majority of the occurrences appear in situations where you should switch your case. It appears the FE created more opportunities to win the grand price.
One more point, if you look at the games that aren’t counted in the DoRD example…
This is the first table of all combinations in DoND. If you take the 4th and 6th rows and change the remaining door to #1 (the winning door). Because if we were playing by Monty Hall Rules that would be the final door you get 66.75% for this simulation.
…and apply Monty Hall rules to it by changing each non-winning door into a winning door. You get 60% to switch as shown above. Using the FE , the prize would always be in play.
Conclusion
It appears there is a major difference between a Forced Elimination vs a Random Elimination.
Don’t believe our results? You can test them out yourselves. We’ve developed a Monty Hall Test Program. You can test our results yourselves using “Monty Hall” rules or “Deal or No Deal” rules. We believe this should be sufficient enough to convince even the most die hard ‘MHDA’ Camper.
