User talk:Lambiam

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The Holy Grail of MHP just got better

Here is a short and elementary and complete solution of the MHP, which actually covers the biased host situation just as well as the usual symmetric case. There is no computation of a conditional probability. All we have to do is to consider two kinds of players: a player who in some situations would stay, and a player who in all situations would switch. We show that both kinds of players are going to end up with a goat with probability at least 1/3. In other words, it's not possible to do better than to get the car with probabillity 2/3. But always switching does give you the car with probability 2/3. Hence always switching achieves the best that you can possibly do.

Suppose all doors are equally likely to hide the car, and you choose Door 1.

If you are planning to stick to Door 1 if offered the choice to switch to Door 2, you'll not get the car if it is behind Door 2. In that case Monty would certainly open Door 3, you'll have the choice between Doors 1 and 2, and you'll keep to Door 1. Chance 1 in 3.

Similarly if you are planning to stick to Door 1 if offered the choice to switch to Door 3, you'll not get the car if it is behind Door 3. Probability 1/3.

If on the other hand you are planning to switch anyway, you'll not get the car if it is behind Door 1. Chance 1 in 3.

Altogether this covers every possible way of playing, and however Monty chooses his door: there's always a chance of at least 1/3 that you'll end up with a goat. This means that there is no way you can do better than getting the car with chance 2/3.

We know that "always switching" guarantees you *exactly* a chance of 2/3 of getting the car. I've just shown you that there is no way this can be improved.

Side remark 1: For those who are interested in conditional probabilities, the previous remarks prove that the conditional probabilities of the location of the car (given you chose door x and the host opened door y) will always be in support of switching. Otherwise, we could improve on the 2/3 overall succcess-chance of always switching, by not switching in a situation indicated by the conditional probability of winning by switching being less than 1/2.

Side remark 2: For those who are worried that I did not talk about randomized strategies (e.g. you toss an unbiased coin to decide whether to switch or stay, when you chose Door 1 and the host opened Door 3) it suffices to remark that you could as well have tossed your coin in advance of the host opening a door. Thus this is the same as choosing a deterministic strategy in advance, by randomization. Since any deterministic strategy gives you a goat with probability at least 1/3, the same is true when you choose one such strategy at random.

Of course 20 text-books in elementary probability theory do MHP in a different way, while the previous analysis is only implicit in recent arXiv.org papers (preprints) of A.V. Gnedin. However as a service to wikipedia editors Richard D. Gill (mathematician) will place this analysis on his university home page so there is at least one reliable source for it. Richard Gill (talk) 13:10, 1 July 2011 (UTC)[reply]

Lewontin's fallacy AFD

Hi. I have a question regarding your AFD comment. I was wondering if you could provide some clarification. Thanks. Guettarda (talk) 18:28, 13 July 2011 (UTC)[reply]

Japanese dialect/s

I was just wondering why you moved Japanese dialects? Considering that there is more than one, is not the pluralized form more accurate? Also why was there no discussion before the move? Colincbn (talk) 01:16, 14 July 2011 (UTC)[reply]

Orphaned non-free image File:HP7part1+2poster.jpg

⚠

Thanks for uploading File:HP7part1+2poster.jpg. The image description page currently specifies that the image is non-free and may only be used on Wikipedia under a claim of fair use. However, the image is currently not used in any articles on Wikipedia. If the image was previously in an article, please go to the article and see why it was removed. You may add it back if you think that that will be useful. However, please note that images for which a replacement could be created are not acceptable for use on Wikipedia (see our policy for non-free media).

Note that any non-free images not used in any articles will be deleted after seven days, as described in the criteria for speedy deletion. Thank you. Sfan00 IMG (talk) 10:57, 18 July 2011 (UTC)[reply]

Article's whose AFD you commented in back at AFD a month later

Since you commented in the AFD for this article last month, I thought you might want to know its back at AFD again this month. [1] Dream Focus 03:20, 22 July 2011 (UTC) \[reply]

Wilma Pang

Please have a look at it as someone seems to be trying to delete it again.