Crosswords0 min ago
anyone fancy a gamble
ok, so i'm on a quiz show and to win the star prize of say 20 quid i have to guess which door its behind. there are three doors and i have two chances. my first guess, door number one. The quiz show host opens up door number three which is empty and gives me the chance to stick with door number 1 or change my mind and select door number two. what should i do?
jim
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Some people at one of my friends work was sorting through this problem by example. Similar to the 52 card illustration - which makes perfect sense - they used like 6 "doors" (or upside down plastic cups) Anyway, the girl who chose the one cup from the six happened to pick the right cup (1 in 6 odds) the very first try. This happened on two consecutive occassions. So in her case, switching would fail the test.
I think the hard thing to get over is the fact that we're not dealing with 52 doors, just 3. But...I stand corrected. When talking with my friend, I first deduced that you stand a 25% better chance at switching in the second round (numbers were dancing in my mind), but then he said that he read you stand a 17% better chance. Anyone care to take a stab?
This is what mathematicians call "counter-intuitive". The way I thought of it was to image 100 doors instead of 3. Suppose you choose door 1, and then the host opens doors 2 to 37 and 39 to 100. He then gives you the option of switching from door 1 to door 38.
This is based on the host's knowledge that the car is behind door 38 (or possibly, door 1). But when the host has opened all the other doors, there is only a 1-in-100 chance that the car is behind door 1 (the one you chose in the first place). There must therefore be a 99-in-100 chance that it is door 38, not a 1-in-2 chance.
The whole point about this is that the host knows where the car is, and you atre therefore not being given a random choice with your option to switch.
the point is not whether the host knows which door the 20 quid is under - although being the host he may well know - it is whether the 20 quid stays behind the same door. if the 20 quid doesn't move then the contestant is stuck with the original probability until he switches doors. the host has no effect on the outcome.
jim
a related q is the Villa Serbelloni problem so called as it almost wrecked a conference held there in 1966
Matthew Mark and Luke are in prison. Two of the three are to die but they dont know which. They know when: the next day. On the last night Mt goes to a gaoler and says one of the other two has to die, it doesnt matter which to me but can you tell me who it is?
The gaoler says Mk. Mt goes to bed much happier because before he asked the question, he had a 2 in 3 chance of dying, but after the answer, it is now 1 in 2.
Is Mt right to feel happier?
From Math ideas in Biology first ed 1969, John Maynard Smith.
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