Author: Rolf Tueschen
Date: 13:21:26 09/25/02
Go up one level in this thread
On September 25, 2002 at 15:06:58, John Merlino wrote: >On September 25, 2002 at 12:38:06, Rolf Tueschen wrote: > >>Please take a look at my revolutionary solution of this confusing problem: >> >>http://hometown.aol.de/rolftueschen/monty.html >> >> >>At first I went into the net and collected all sort of data for my page. I >>wanted to show how important methods and methodology are for science and also >>statistics. In special the exact defining of the terms. >> >>Then suddenly I had the inspiration and in a few minutes whitewashed a million >>people who as pupils, students or even professors let them be proved wrong by >>Marilyn vos Savant who has an IQ of 228. For decades now the Monty Hall Problem >>is taken as example for conditioned probability, which is wrong! >> >>Hope you enjoy my revelations. Please tell me if you want to comment. >> >>Rolf Tueschen > >You are incorrect. The problem AS STATED gives a 2/3 probability of being >correct if you switch. The "host" has nothing to do with it, because he/she MUST >choose a door to open. > >The REAL Monty Hall, IIRC, commented on this problem and he stated that this >problem did NOT mimic the actual event on the "Let's Make a Deal" game show. On >the show, he had a CHOICE of whether or not to open a door, meaning that, >regardless of whether or not a door was opened, the probability of being correct >was always 1/3. I also believe that vos Savant later commented on this variation >as well. > >However, in this problem, a door MUST be opened, and therefore the chances of >being correct are 2/3. This is simple to simulate, and it is a very common >problem in beginning programming courses. Here is the output from my version: > >Cadillac in 1, Picked 1, Showed 0, Switched to 2, GOAT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 0, Picked 0, Showed 1, Switched to 2, GOAT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 0, Picked 0, Showed 1, Switched to 2, GOAT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 1, Picked 1, Showed 0, Switched to 2, GOAT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 1, Picked 1, Showed 0, Switched to 2, GOAT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 2, Picked 1, Showed 0, Switched to 2, CORRECT! >Cadillac in 2, Picked 1, Showed 0, Switched to 2, CORRECT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 2, Picked 1, Showed 0, Switched to 2, CORRECT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 2, Picked 0, Showed 1, Switched to 2, CORRECT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 0, Picked 0, Showed 1, Switched to 2, GOAT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 2, Picked 1, Showed 0, Switched to 2, CORRECT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 1, Picked 1, Showed 0, Switched to 2, GOAT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 1, Picked 1, Showed 0, Switched to 2, GOAT! >Cadillac in 2, Picked 1, Showed 0, Switched to 2, CORRECT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 2, Picked 1, Showed 0, Switched to 2, CORRECT! >Cadillac in 2, Picked 0, Showed 1, Switched to 2, CORRECT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 1, Picked 1, Showed 0, Switched to 2, GOAT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 2, Picked 0, Showed 1, Switched to 2, CORRECT! >Cadillac in 0, Picked 0, Showed 1, Switched to 2, GOAT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 0, Picked 0, Showed 1, Switched to 2, GOAT! >Cadillac in 2, Picked 1, Showed 0, Switched to 2, CORRECT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 1, Picked 1, Showed 0, Switched to 2, GOAT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 2, Picked 0, Showed 1, Switched to 2, CORRECT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 1, Picked 1, Showed 0, Switched to 2, GOAT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 2, Picked 1, Showed 0, Switched to 2, CORRECT! >Cadillac in 0, Picked 0, Showed 1, Switched to 2, GOAT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 2, Picked 0, Showed 1, Switched to 2, CORRECT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 1, Picked 1, Showed 0, Switched to 2, GOAT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 0, Picked 0, Showed 1, Switched to 2, GOAT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 2, Picked 2, Showed 0, Switched to 1, GOAT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 2, Picked 0, Showed 1, Switched to 2, CORRECT! >Cadillac in 1, Picked 2, Showed 0, Switched to 1, CORRECT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 0, Picked 0, Showed 1, Switched to 2, GOAT! >Cadillac in 0, Picked 2, Showed 1, Switched to 0, CORRECT! >Cadillac in 1, Picked 0, Showed 2, Switched to 1, CORRECT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 0, Picked 1, Showed 2, Switched to 0, CORRECT! >Cadillac in 2, Picked 0, Showed 1, Switched to 2, CORRECT! > >71 out of 100 times was correct when switching > >Running it five more times resulted in 68, 68, 66, 64 and 69. Obviously 2/3 over >the long haul. > >Basically, it boils down to the only time you lose is if you initially chose >correctly, which has a probability of 1/3. Therefore, the winning probability is >2/3. > >jm Thanks for the energy you gave me as a present. I knew that the whole page is a bit long, so I can only tell you that all points are well mentioned there. I quoted the original Monty Hall as well. And you are also quite right with your simulation. I found quite a few automats on the net ready for download. But let us find out together what's up here. Of course you are right, mathematically etc. My special point, the whole rejection of the whole maths well copied from one to another on the net, but never with the basic reflections, is that the application of that conditioned probability is wrong in the case of just one single unique event. Probability is something, as I showed in extensive quotings a question of n>1. And there is no effect on learning from the candidate's choosing a door but NOT opening it of course and then the host opening a wrong door in addition. I correct. This is no increase in learning above the chance level. It increases from 1/3 to 1/2, but that is for both doors still remaining. Now all what I said was that the 2/3 was a delusion, because it can't be applied for the unique event. But if you take 100 people in a group, I agree that then a majority will win with the switch option. But as you know a majority is not everyone. And if some experts insist (on my page) that it was asked for the chances in the first question I must say that for the single individual it's a difference if he will surely win in his (unique) event or if he knows that 66 of 100 would win. He might _not_ win nevertheless. So, as I remarked, and that is crucial, the exact definition of chances and advantage and such is very important. In sum. If I had the same question outside the TV show,say in medicine, and I had the choice here between two posibilities, and the doctor would tell me that I had a 66% chance to survive with method B, I would _not_ decide between A and B but would do all I could to get more information. For the single case 66% overall doesn't mean clarity. For such n=1 cases it's more a psychological problem than a mathematical. And all what this example should prove was the importance of exact difinitions and care for the control of your variables in statistics. Rolf Tueschen
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