Before You Do The Math, Do The Strategy
Let's now get down to the more serious side of life and its confusion of strategies and look at a famous example known as The Monty Hall problem, a mathematical probability puzzle based on the American television game show Let's Make a Deal:
In the Monty Hall conundrum scenario, where you, as a game show contestant, have picked one out of three doors with no prior knowledge of which one conceals the prize, your odds of being right are approx. one in three. Then if the host, Monty, who KNOWS (even if you don't know he knows) what's behind the doors, opens one of the two you didn't pick, and it turns out to be also the wrong door to the prize, should you, when now given that option by Monty, switch your pick to the third door?
Most say that the odds are no longer 1 in 3 but are now 50-50, meaning there's no "probability" difference between the remaining picks. But no less a person than the brainy Marilyn vos Savant has advised her readers that the player should switch, after which math mavens said she was nuts.
But to me the key element here is that Monty, when he opened and eliminated one door, obviously knew in advance which door to eliminate - so luck had nothing to do with his move, and seemingly changed nothing as far as the overall odds were concerned. But Monty's move did in fact change the odds for you, the player, who in effect have been given two choices rather than one, and this was no doubt the intent all along.
Except your new circumstances also tell us that while you had a one in three chance initially, it now appears that you have one in two chances to get it right. Causing many, if not you in particular, to feel the choice to move again or not move again will on average have the same result. So if you decide to do nothing, you have in effect made the second choice to not take a chance that you already took to begin with. Not pushing your luck, so to speak.
But what you won't likely consider is that in fact there's still a 2/3rds (approx.) chance you CHOSE WRONG initially. So one more choice (which has now become simply a switch) in this situation ON AVERAGE will pay off more often than not. It would now seem you are actually exchanging 1 in 3 initial odds for (approx.) 2 in 3 reconstituted odds of CHOOSING RIGHT if you switch. Because it also appears that your chances of twice choosing wrong have been reduced to 1 in 3!
And even more to the point, if given the "second move" option WITHOUT THE REVEAL you would NOT have had the impetus to take that second step. You would not have had that clue as to which way to move when taking it. And now you have both the clue to the move and favorable odds for making it!
Which, as I hinted earlier, are opportunities you likely don't know you have. The kicker being that, paradoxically, that same clue could have given you the idea that BECAUSE of the "reveal" you would not improve your odds at all by moving. Which would appear to have been Monty's tactical intention as well. So if you were thereby manipulated into thinking this latter view was the correct inference to be drawn, you would have been had.
And as far as I've read up on this subject, the most perspicacious have observed it's because of Monty knowing what was behind each door that the conundrum not only existed but was imminently solvable. Which I'm now of the opinion is wrong as well. Because the key to its ultimate solution was not simply that Monty KNEW this, but that he REVEALED that knowledge to the player.
So that while all of this is ordinarily viewed as a math problem, it's actually, in my view, a strategic conundrum with its mathematical aspects concealed (as strategies are wont to do). It is the strategic plan here that makes use of math while at the same time being structured to deceive both players and observers in ways that can't be constructed or adequately explained by that very mathematical process.
Do I really think any of the above makes sense? Well, since most everyone now seems to agree with Marilyn vos Savant that one actually should switch in this situation, this effort of reasoning "backwards" from there to sort things out strategically says (to me at least) all this is surely part of the how and why such a puzzle was formed. Because my premise was and still is that all conundrums have at bottom a strategically deceptive purpose.
(And from what I've since read, those Let's Make a Deal players who actually switched benefitted from the switch on a 2 to 1 basis, except there seems to have been no attempt to determine the odds involving how many would or would not take advantage of that opportunity.)