This post presumes you read the other 2 Monty Hall posts. Click on them to read them then return here.
Lock The Gates!
Just when we thought we understood everything about the Monty Hall Problem our world gets turned upside-down by the
Monty Hall Prophecy
Once again, let’s recap the problem:
We choose door #1, Monty shows us an empty door#3, and then asks us if we want to switch to door#2.
What should we do?
1. Should we stick with our original choice of door No. 1?
2. Should we switch to door No. 2?
3. Or… Once again, as a whole lotta people believe, WHAT DIFFERENCE DOES IT MAKE whether we stick with door #1 or whether we switch to door #2. They claim that all we know is that one door has the prize and one doesn’t so we would have a 50/50 chance of winning either way.
This is the part of the problem we will now focus on but this time from a different perspective being pushed by many liberal/progressive “FiftyFifty-ers” whom we’ll call the “5050s”
As we have proven multiple times, if we want to maximize our odds of winning the prize we should switch to door No. 2. By this time nobody but a 5050er should dispute that. However, 5050s still believe that ”All you know is that one door has the prize and one doesn’t so you would have a 50/50 chance of winning either way.” Interestingly,
The 5050s can seemingly fulfill their misguided prophecy
in a roundabout way… Let’s see how!
So again, they choose door #1, Monty shows them an empty door#3, and then asks them if they want to switch to door#2. At this point there is, however, a ploy that 5050s will use “prove” their prophecy that it doesn’t matter whether we stick with door#1 or switch to door#2.
The 5050s say that we should flip a coin and
if it’s HEADS we stick and if it’s TAILS we switch to door#2.
What could be more “Doesn’tMatterish” than that!? What could be more “Fifty-Fiftyish” than that!? And what will our odds be of picking the door with the prize if we follow this coin flipping strategy? Let’s pause for a minute or so to ponder this… tick tick tick…
At this point, we could think about the problem and then compute our odds. And we could also (and easily) run our own simulation dozens of times using a deck of cards and a coin to confirm or disprove the theory. Or we could also enhance the MontyHall Simulation C# program (which I did) to have it make the “door choice” based on a coin flip. And let me tell you, it ain’t easy to get a computer program to flip a coin!! Anyway, the enhanced Monty Hall Simulator program executed billions of LMD simulations using the “coin flip rules.” In all 3 cases (thinking, cards and coins, and computer) the results confirmed the 5050s Prophecy every time! That is:
When we flip a coin to decide whether or not to switch doors
then indeed our odds of winning the prize are 50/50 !!!!!
This 5050s coin flipping strategy is what I like to call the Monty Hall Prophecy because it’s a self-fulfilling prophecy. And we have to admit that the term “Monty Hall Prophecy” has a nice ring to it, playing as it does on the turn of the phrase Self-fulfilling Prophecy. But Self-fulfilling or not, it’s still interesting to analyze.
Anyway, the 5050s will claim that flipping a coin will confirm the ”Doesn’tMatter Prophecy” because, again, what is more “Doesn’tMatterish” than flipping a coin!? So lo and behold, what happens when we do this? Our “Doesn’t Matter” prophecy and associated 50/50 odds of winning the prize are confirmed!!
But how can that be!?
I thought we had previously proven that
if we always switched doors we would have a 2/3 chance of winning?
But now it appears that it “doesn’t matter” because
if we choose between the 2 doors by flipping a coin
then our odds of winning are indeed 50/50
which is quintessential “doesn’t matter!”
What’s going on!?
Before we end, just below is a screen print of the Monty Hall Simulator program after it was enhanced to allow for the “Coin Flip” door selection option. Note the use of Radio Buttons which C# in Visual Studio makes so very easy and simple to use in your app. Visual Studio really is a great product!
Ok. Now we are done. I think we’ve pretty much exhausted the whole Monty Hall thing. Now we can finally leave…
Open The Gates!