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Message: <blockquote data-quote="Ovinomancer" data-source="post: 7270116" data-attributes="member: 16814">No, and this:Is pretty close.The 3 doors version of this problem is the simplest, but it extends.  I'll try to present the 4 sites version of it.You are initially presented with 4 choices, only one of which is correct.  You have no other information on the choices at this time. You chance of naively picking the correct choice is 1:4 (assuming a fair contest, as always).Now it get interesting, because the Monty Haul stand in is going to provide more information before we see if our choice is correct, but, as 26 notes, that information is privileged information that is provided knowing the full truth of which choice is correct.  Monte will now provide information about one of the choices you didn't pick, but will always, always, always show you a wrong choice.  This is important -- Monte's information is not random at all, but must show an incorrect choice from those that you did not pick.So, to back up a moment, your chance of picking the correct site at first is 1:4.  This means that there's a 3:4 chance that you picked the incorrect site and the correct site is one of the 3 remaining sites.  When Monte tells that one of the unpicked sites is wrong, he's providing much more information that is first apparent.  In the likely event that you picked incorrectly, you now have one fewer incorrect other choices.  In other words, the likelihood that the correct site is one of the choices you didn't pick has changes from 3:4 to 3:8 (as twosix points out).  The why is more apparent if you break it down a bit.Odds your first choice is correct (1 pick from 4) is 1:4.  This never changes because it was picked when you had no information about the other doors.  The reveal of one other incorrect site doesn't alter these odds because you already knew there were at least 2 wrong sites among the other picks (the case where you picked the correct site first and the case where you didn't), so this information doesn't change the odds of your first pick being correct.  It does, however, change the odds that one of the 2 remaining unpicked sites is correct.  In the pre-reveal information, the odds that each site is the correct site is 1:4.  This adds up to a 4:4 total odds, or a probability of 1 that the correct site is among those four choices.  But when Monte reveals one incorrect site, this changes.  The odds you picked correctly to begin with are still 1:4 (see above for why), but now there are only two possible choices left instead of 3.  The probabilities must still add up to 1, and we have the 1:4, so the remaining 2 choices have to split among the original 3:4 probabilities, so we get 3:8 chance that each of the remaining two choices are the correct site.  3:8 + 3:8 + 1:4 is 4:4, so we've satisfied the 1 probability requirement.That's how the Monte Hall problem works, generally.  By providing new information after the choice but before the reveal, the odds of success are shifted.  The more choices available, the slighter the <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":)" title="Smile    :)"  data-smilie="1"data-shortname=":)" /><img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":)" title="Smile    :)"  data-smilie="1"data-shortname=":)" /><img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":)" title="Smile    :)"  data-smilie="1"data-shortname=":)" /><img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":)" title="Smile    :)"  data-smilie="1"data-shortname=":)" /> (at 3 choices it goes from 1:3 to 2:3, for instance).To give a slightly different example, let's look at a more extreme reveal.  If you have 15 sites to pick from, and, after you pick, Monte looks at the remaining 14 sites and then reveals 13 of them that are wrong, leaving only one site, and asks if you want to switch, it's really apparent that switching is far more likely to pick the correct site.  While this is extreme, it does more clearly show the effect that the additional information has.  Even if Monte only told you about 1 of the remaining 14 sites, the same effect, at a much smaller impact, would be there and you'd still, on average, be better off switching your guess.</blockquote>

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