My proof of the "Let's Make a Deal" Problem
On the TV Show "Let's Make a Deal" a contestant had to pick one of three closed curtains. Behind one was a big jackpot but a much lesser prize behind the other two. After the contestant picked a curtain number closed curtain (1, 2 or 3), the MC would show them another curtain that had one of the booby prizes. He then asked the contestant if they would like to switch their choice.
The logic puzzle is "Does it make any difference if the contestant switches his guess of which curtain he/she wants?" In other words, Does the chances of the contestant winning change if he switches his guess?
The surprising, and somewhat counterintuitive, answer is "Yes, the contestant wins twice as often if he/she switches his guess to the 'other' non-shown curtain."
There's many proofs. Here's one I think is simple: When the contestant first selects a curtain, there are 3 curtains from which to choose and only one is the correct one. Now there's two cases:
1) Suppose he initially picked the correct one (1 chance in 3) - then switching to the other curtain will hurt him as he then gets the booby prize, and
2) Suppose he initially picked one of the two booby prize curtains (2 chances in 3) - but then he's already been shown the other booby prize curtain (which he won't select when he switches), so when he switches he will be switching to only remaining, correct curtain.
So 2 times out of 3 (66.67%) switching his/her initial pick to the other non-shown curtain will help him/her, so the contestant should switch. QED
Here is a link to some of the flak Marilyn Vos Savant received when she properly tried to explain this in her "Ask Marilyn" column. For more, just "Google" this issue.
On the TV Show "Let's Make a Deal" a contestant had to pick one of three closed curtains. Behind one was a big jackpot but a much lesser prize behind the other two. After the contestant picked a curtain number closed curtain (1, 2 or 3), the MC would show them another curtain that had one of the booby prizes. He then asked the contestant if they would like to switch their choice.
The logic puzzle is "Does it make any difference if the contestant switches his guess of which curtain he/she wants?" In other words, Does the chances of the contestant winning change if he switches his guess?
The surprising, and somewhat counterintuitive, answer is "Yes, the contestant wins twice as often if he/she switches his guess to the 'other' non-shown curtain."
There's many proofs. Here's one I think is simple: When the contestant first selects a curtain, there are 3 curtains from which to choose and only one is the correct one. Now there's two cases:
1) Suppose he initially picked the correct one (1 chance in 3) - then switching to the other curtain will hurt him as he then gets the booby prize, and
2) Suppose he initially picked one of the two booby prize curtains (2 chances in 3) - but then he's already been shown the other booby prize curtain (which he won't select when he switches), so when he switches he will be switching to only remaining, correct curtain.
So 2 times out of 3 (66.67%) switching his/her initial pick to the other non-shown curtain will help him/her, so the contestant should switch. QED
Here is a link to some of the flak Marilyn Vos Savant received when she properly tried to explain this in her "Ask Marilyn" column. For more, just "Google" this issue.