Of the remaining 66.6% of the time there is no benefit in switching,each contestant is likely to win 50% of the time, or overall each has a 33.3% of winning.

The increased likelihood of winning by switching (in the regular game) , is because Monty can NEVER open the car door. If the game is changed such that Monty doesn’t know where the car is and accidently opens a goat door, then there is no benefit in switching.

]]>Let’s say there are two contestants in the game instead of one. Each contestant may pick any door he or she prefers. If they pick the same door and stick with their choices, each contestant would win whatever prize is behind that door. However, one contestant chooses door # 2, and the other chooses door #3. Monty then reveals that the prize behind door #1 is a zonk. Now he gives each constestant the option of switching doors.

According to the theory that Barbara lays out, each contestant has a two-thirds chance of winning the big prize by swaaping doors. How can that be?

I maiintain that this conundrum exists even when only one contestant is playing the game. If the contestant had chosen door #2 initially, he or she would have greater odds by switching to door #3. But, if that contestant had chosen door #3 initially, he or she would be more likely to win by switching to door #2.

]]>It’s important to bear in mind Monty is actively trying to make the contestant lose (I assume). In other words if the person picks door number 1, and the car is actually behind door number 1, I suspect Monty is going to pull out all the stops to try to get you to switch from door 1, including body language, emphasis on certain words, and even revealing the fourth door/curtain they occasionally sprung on the contestant, just to increase the odds further against the contestant sticking with the door they chose.

There’s a problem with the usual setup of the problem though, because all the times I watched the show, I don’t remember there being a “super” prize and then two joke prizes — there was usually a “great” prize, a “joke/useless” prize, and a “nice” prize as a consolation. If the great prize was a car worth $2000 in 1970s dollars, there’d usually be a prize of a new washer and dryer, or a trip somewhere, worth maybe $300-400. I doubt Monty really tried to steer people to the joke prize, I imagine he tried to steer them to the medium, consolation prize — they leave happy, the sponsor is happy, Monty doesn’t go to bed with a guilty conscience, etc. At least if I was the producer of the show that’s what I’d try to do.

Personally I like Ryan’s use of the cards, but I think 52 is more than you need to make the point. I think 10 would be sufficient. Even only one or two additional ones change the equation enough to make the solution more apparent.

Now here’s a question to ponder: do folks think Monty, and the show’s producers and sponsors understood about the 1/3-2/3 different in probability? Or did they go through the run of the show blissfully unaware of the math?

]]>After you choose a door, Monty says “I could open a door with a zonk behind it, because there’s one car and two zonks behind the doors, and offer you the chance to switch. But I’m lazy, and the zonks are worthless, so I’ll just give you the chance to trade your one door for BOTH of the remaining two doors.”

Now it’s easier to see that switching wins.

]]>Clearly you do, as the chance of your original door being correct was 1/100. The odds were in favour of you being wrong and therefore are now in favour of being right if you switch.

]]>But I love the 52 card demo. I think that makes a lot of sense and you could even try it with less cards until they understand.

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