Question

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 other 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 take the switch?

Answer 1

Yes you switch. This is a famous problem called the Monty Hall Problem.

Answer 2

good job its almost too recognizable now i wanted to test the people on this site to know if they are actually doing their job! can you prove statistically why it is in your best interest to switch?

Answer 3

Oh lol interesting :P Ya actually a really short solution:
Consider \(A\)=opening a door without the prize
\(B\) opening a door with a prize after switching

then \(P(B) = P(B|A)P(A) + P(B|\bar{A})P(\bar{A})\) by the law of total probability

\(P(B) = 0(1/3) + 1(2/3) = 2/3\)

Answer 4

oops that should be A, opening door with the prize...

Answer 5

ok awesome thanks!!

Answer 6

I think this is one of the shortest solutions I can think of. I know though when first seeing this, it is more intuitive considering the possibilities on a case-by-case basis, or even doing a tree diagram (which comes up the the same idea)

Answer 7

well and just using percentages. By sticking true to switching the door every single time, you have a 66% chance of choosing wrong the first time and by switching, you will be switching to the car 66% of the time

Answer 8

yup :)