Posted by TheFallen on October 17, 2004

In Reply to: The Monty Hall puzzle posted by Fred on October 17, 2004

: : : : : : : : : : : : : This was referenced by someone in a post slightly lower down and is a neat term to describe the scientific truth that past events do not affect future probabilities - so if you've spun a coin 19 times and it's come down heads each time, there's still an exact 50/50 chance that it'll come down heads or tails on the 20th spin... presuming the coin's perfectly balanced and that it doesn't land on its edge.

: : : : : : : : : : : : : This reminds me of the famous TV game show goats paradox, which I'll outline briefly for those who don't know it. You get to the final of a TV game show - there's just you left. You're presented with three closed doors by the host, who tells you that behid one of the doors is a brand new Ferrari, but behind the other two there are goats... so three doors, one sportscar and two goats. You are invited to pick a closed door, so you do. The host, who obviously knows in advance what's behind each door, then decides to open one of the two doors that you didn't select to reveal a goat - he does this every week. He then asks you whether you want to stick with the original door you picked, or whether you want to swap to the remaining closed door. The simple question is - if you want to maximise your chances of getting the Ferrari, would it be best to stick with your original choice, should you swap, or does it make no difference at all?

: : : : : : : : : : : : : (Apologies if you've all heard this before).

: : : : : : : : : : : : I'm so bad at this stuff but surely it makes no difference. If it does, please speak slowly and in plain English. Assume nothing in your answer.

: : : : : : : : : : : : Would it be so bad to get a goat? It'd be far worse to get a rat or snakes.

: : : : : : : : : : : It'd be a little more disappointing than winning the Ferrari though. Camel's vote is in - she maintains that it can make no difference whether she sticks to her door or swaps her choice to the other door... any other opinions?

: : : : : : : : : : Taxes would be much less on a goat. Most of us could not afford the taxes on a Ferrari.

: : : : : : : : : I think I'm right in saying that prizes won on game shows are not taxable - however it'd be far cheaper to insure the goat, that I grant you.

: : : : : : : : It's better to switch. I don't remember the proof, though. Actually, I'd rather have a goat than a Ferrari, practical considerations aside, like where to graze it.

: : : : : : : I thought we couldn't win the goat any more?

: : : : : : : FYI prizes aren't taxed in the UK. In the US *everything* is taxed. So when you are figuring out what to do with your NY lottery millions, remember to count on about half. And always take the lump sum.

: : : : : : The redoubtable Ms. Berg is correct - changing your mind and switching doors increases your chances of winning the car to exactly 2 to 1.

: : : : : : "How?" I hear you muttering "There's just two closed doors left, with one sportscar and one goat... it has to be a half chance."

: : : : : : Well, no it doesn't and here's why. Think about the first choice you made when there were three closed doors. You had a 2 out of 3 chance of choosing a door with a goat behind it. Therefore, given that the host decided to reveal a goat behind one of the closed doors that you *had not* picked, on two out of three occasions the host had to be showing you *the other goat*. Only if you were unlucky enough to have hit on the door with the sportscar behind it on your first choice would the host have the luxury of revealing *either goat*. Therefore, you should always switch doors.

: : : : : : (If you either don't understand this or refuse to believe it, try it for yourself with three playing cards - always be a switcher and see what happens).

: : : : : Thanks for clearing that up. I had a feeling that untaxable game show winnings was not a reality in the US.

: : : : I don't really understand why but I'm prepared to try it next time I'm in the situation.

: : : I'm a little confused. (I'm usually that way, so don't be alarmed.) But you say that the host "decided to reveal a goat" ... but, earlier, you said the host does this EVERY week. No "decision" involved. It will always be possible for the host to reveal a goat no matter what you pick, so this little ritual has no bearing on what you chose. (Still sounds like 50-50 to me.) Please clarify.

: : Badly expressed by me. It's not a decision and as you rightly say, the host invariably reveals a goat. However, given that you have a 2/3rds chance of picking a goat at the first door selection stage, and given that the host will never at that point reveal what's behind the door you first choose... then 2/3rds of the time after your first selection, the host is effectively compelled to show you *the other goat*. It's only if you happen to have picked the door with the car on your first selection, that the host can reveal *either* of the two goats. Knowing this, then 2/3rds of the time you should switch your choice.

: : It's a little tricky to express in words, and I am no statistician, but in this specific instance, it at least superficially appears that past events can affect future probabilities.

: : Take heart though. I've laid out this problem to many people before, and it's usually those of a mathematical persuasion who get most confused.

: Joseph Y. Halpern discusses the "Monty Hall puzzle" (the one-car-two-goats puzzle) in his _Reasoning about Uncertainty_ (2003, MIT Press).

: I gather from his discussion that there is not total agreement among the experts about how to handle the puzzle. He refers the reader to F. Mosteller (Fifty Challenging Problems in Probability with Solutions, 1965, Addison-Wesley), M. von Savant (Sept. 9, 1990, Parade Magazine) and J.P. Morgan et al. (1991, The American Statistician).

Thanks for this. This puzzle first appeared in 1990 and caused a storm in the mathematical world, with many well-qualified people initially refusing to believe the results. If still unconvinced by my explanation above, try it yourself at

http://people.hofstra.edu/staff/steven_r_costenoble/MontyHall/MontyHall.html

...which features pigs, not goats, but I don't think this is likely to cause any measurable effect on probabilities.

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**Lewis***18/October/04*- Counter-intuition
**TheFallen***18/October/04*

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