The problem goes as follows.
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 he knows 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?and the same in a dis-ambiguated version to fix assumptions is as follows.
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. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you “Do you want to switch to Door Number 2?” Is it to your advantage to change your choice?If you have not come across this puzzle, try it. What would have been your answer? (No, it doesn't really matter, because there is an equal chance (or) Yes, I'll switch, what the heck (or) Leave me alone, my brain hurts) The solution has been discussed umpteen times, the problem statement reworded, variant behaviors discussed at length, proof in multiple forms, but this problem at times still confuses me because it is counter-intuitive to what we feel, think and assume about the problem. Probability and its conditionalities. Hmmm... (Somehow this problem reminds me of how Surya makes a scapegoat of Laila in glib talk + gambling in the Tamil movie Pithamagan :-))
Another interesting aspect of math theory is paradoxes. One of the famous ones is the liar paradox.
Epimenides was a Cretan who made one immortal statement: "All Cretans are liars."How do you perceive the above statement? For me, it is like a hung process in Unix. :-)
And as regards perceptions and illusions, what else, but M. C. Escher's pictures for the visual paradoxes? Reality turned on its head with clever proportions. What an artist!!!
WaterfallAll this reminds me of school/college days when we used to prove 0=1, 1=2 etc for fun using seemingly correct Algebraic formulations or differentiation rules.
Copyright M. C. Escher
Official M. C. Escher website - http://www.mcescher.com
All copyrights acknowledged
Wish if only all Math were simple and as interesting without the tons of seemingly dry and drab theory that accompany the derivations...
Have a wonderful weekend.
Source acknowledgement: Wikipedia, Forbes. All copyrights acknowledged.