Having only taken math through advanced calculus and differential equations, none of the examples make much sense to me.
But there is the "Monty Hall problem", where many mathematicians believe(d?) the incorrect result.
But there is the "Monty Hall problem", where many mathematicians believe(d?) the incorrect result.
Then, someone springs the probability for the problem they poorly described and feels proud they fooled us all!
I _hate_ the Monty Hall problem for this reason. It’s not a function of statistics but rather a function of an author’s technical writing ability.
I don't think it is just poor wording.
And the idea that probabilities can shift based on Monty's knowing where the prize is certainly not intuitive, in part because many of us do not intuit accurately about probability.
> Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
I would just point out the obvious that the game show host will never pick the door with the car.
You could say that the wording given kind of implies the always, but it's not completely clear. So, the claim that it's almost always worded poorly is reasonable.
Left unspecified: does the host _always_ follow this exact procedure unconditionally? Does the player know this to be a fact?
Is the hosts algorithm exacly:
1) let the player choose a door 2) open _a_ door that was not the players choice that the host knows contains a goat 3) let the player change choice if they'd like
If it's not, then depending on what the host's algorithm is, then it's not even a question of probability or the odds might differ. Also depending on what the player knows or doesn't it may stop being a question of probability.
"vos Savant herself was flooded by a surge of disbelief. She received more than 10,000 letters from readers of her column, the vast majority of whom were absolutely convinced that she was wrong. Among them were many PhDs and a strikingly large number of mathematicians. One understanding mathematician kindly offered vos Savant some comforting words: “You made a mistake, but look at the positive side. If all those PhDs were wrong, the country would be in some very serious trouble.”"