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10-26-2011, 07:24 PM   #1981
grinder

Join Date: Dec 2006
Posts: 420
Re: The Official Math/Physics/Whatever Homework questions thread

Quote:
 Originally Posted by checktheriver The computation is correct, P(A|B^c)=1/2. However, what we are really interested in is P(A|Bg) (and P(A|Cg) ), where Bg is the event "the guard tells him B will be set free" (resp. Cg for C). Notice that these events are not the same : Bg is always a subset of B^c (same for C), but you cannot have both Bg=B^c and Cg=C^c, because on the intersection of B^c and C^c (i.e. A), the guard must choose between saying B or C. The probabilities P(A|Bg) and P(A|Cg) will depend on the guard's "strategy", i.e. what he says when both B and C will be set free. For instance, you can try to show that : - if he picks 50% B or C, then P(A|Bg)=P(A|Cg)=1/3, - if he picks B, then P(A|Bg)=1/2 and P(A|Cg)=0.
My answer was writtin thinking they were the same, B^c was the event "the guard tells him B will be set free". Its been a while since ive done anything to do with sets and Id forgotten that ^c was common notation for complements.

Also I should have been clearer that I was assuming the guard's "strategy" was to show no preference to picking B or C. Perhaps this was unjustified, but it seemed trivially obvious that a guard with a bias giving you information is going to affect the probability of your own execution.

10-27-2011, 12:19 AM   #1982
journeyman

Join Date: Aug 2007
Posts: 272
Re: "Monty Hall" style problem

Quote:
Originally Posted by WAtR
Here is the problem:

Quote:
 Three prisoners are informed by their jailer that one of them has been chosen at random to be executed and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information because he already knows that at least one of the two will go free. The jailer refuses to answer this question, pointing out that if A knew which of his fellow prisoners were to be set free, then his own probability of being executed would rise from 1/3 to 1/2 because he would then be one of two prisoners. What do you think of the jailer's reasoning?
This sounds a lot like my favorite variant of the Monty Hall problem, which has the same setup as usual with the one caveat that Monty forgot to take his medication and can't remember which door has the prize behind it. So you pick Door A. In desperation, he flips Door B to reveal... nothing! Mopping sweat from his brow, he then asks whether or not you'd like to switch. Should you?

It's kind of fun to pose this question to people who have just grasped the solution to the original version of the problem. They never get it right.

 10-27-2011, 03:59 PM #1983 old hand     Join Date: Mar 2009 Location: Yes Posts: 1,332 Mathematical Proof f : X → Y is injective if and only if, given any functions g, h : U → X, whenever f ∘ g = f ∘ h, then g = h Again, I need to prove the above. I've been looking for a few hours now w/o success Someone please help me out.
 10-27-2011, 05:47 PM #1984 adept     Join Date: Jul 2008 Location: Oh **** I'm dead !! Posts: 955 Re: Mathematical Proof Isn't that just an extended definition ?
 10-27-2011, 06:14 PM #1985 Carpal \'Tunnel     Join Date: Sep 2004 Location: farther back Posts: 6,604 Re: Mathematical Proof Should be in homework thread? ⇒ Fix g,h: U → X. Suppose that for some u ∈ U, we have f(g(u)) = f(h(u)). Because f is injective, g(u) = h(u). In particular, if f(g(u)) = f(h(u)) for all u ∈ U, then g ≡ h. ⇐ Suppose f(x1) = f(x2) for x1, x2 ∈ X. We now define two functions g,h: U → X. For all u ∈ U, g(u) = x1. For all u ∈ U, h(u) = x2.Thus f(g(u)) = f(h(u)) for all all u ∈ U. By assumption, this requires g ≡ h; which means x1 = x2. This proves f is injective.
10-27-2011, 07:44 PM   #1986
Pooh-Bah

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Re: Mathematical Proof

Quote:
 Originally Posted by Dasq1306 Isn't that just an extended definition ?
It's the categorical definition of an injective morphism (no points/elements involved, just other morphisms).

 10-28-2011, 01:15 PM #1987 old hand     Join Date: Mar 2009 Location: Yes Posts: 1,332 Re: The Official Math/Physics/Whatever Homework questions thread Great, thanks for the responses
 10-30-2011, 03:56 PM #1988 Pooh-Bah     Join Date: May 2005 Posts: 5,653 Re: The Official Math/Physics/Whatever Homework questions thread really not quite a homework problem but... I'm in engineering and taking cal II, doing ok in the class, but I feel like any of the conceptual advanced applied questions I don't have a good sense of how to relate/put it into a mathematical formulation. any tips on how to make it relevant? or how to learn to better apply it to problem solving?
 11-03-2011, 09:47 PM #1989 veteran     Join Date: Jan 2007 Location: PDX Oregon Posts: 2,466 Re: The Official Math/Physics/Whatever Homework questions thread Can someone remind me how to factor this kind of trinomial ? x^2-xy-30y^2
11-03-2011, 10:06 PM   #1990
banned

Join Date: Feb 2007
Posts: 17,356
Re: Mathematical Proof

Quote:
 Originally Posted by lastcardcharlie It's the categorical definition of an injective morphism (no points/elements involved, just other morphisms).
More precisely, of a monomorphism. In Sets, these would be the injective functions. (Injective has a different meaning in other settings; in particular it refers to having an extension property w.r.t. monomorphisms.)

11-03-2011, 10:11 PM   #1991
Carpal \'Tunnel

Join Date: Mar 2007
Location: Redoubling with gusto
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Quote:
 Originally Posted by Acemanhattan Can someone remind me how to factor this kind of trinomial ? x^2-xy-30y^2
same idea as factoring a quadratic in one variable.

how do you factor x^2 - x - 30?

11-03-2011, 10:21 PM   #1992
veteran

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Location: PDX Oregon
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Re: The Official Math/Physics/Whatever Homework questions thread

Quote:
 Originally Posted by Wyman same idea as factoring a quadratic in one variable. how do you factor x^2 - x - 30?
(x-6) (x+5) i believe

I see now. Thanks

 11-04-2011, 01:24 PM #1993 grinder   Join Date: Dec 2010 Location: T.Dot Posts: 493 Re: The Official Math/Physics/Whatever Homework questions thread Expected Value question: The normal weekly demand of a certain perishable product sold by AHOLE Inc is given by the following distribution: Demand Probability 8 0.1 9 0.2 10 0.1 11 0.4 12 0.2 The product cost AHOLE \$5 each. the product sells for \$11 each. if not sold by the end of the week, the leftover units must be scrapped. The supplier only has 10 or 11 units available for AHOLE to purchase. how many would you recomend AHOLE to purchase based on expected profit? (10 or 11) What is the maximum expected profit? Im stuck because if AHOLE can only purchase 10 items, does that factor the demand of 11 and 12 out of the question? if so, would this effect the probability of the other (8 9 10) demand since these would be eliminated? can anyone try and lead me into the right direction? THanks for your help
 11-04-2011, 01:27 PM #1994 veteran   Join Date: Feb 2011 Posts: 2,833 Re: The Official Math/Physics/Whatever Homework questions thread If the demand were 11, but AHOLE only had 10 units, how many would they sell? How much profit or revenue would they get. If the demand were 12? Now, if AHOLE had 11 units, ...
11-04-2011, 03:14 PM   #1995
grinder

Join Date: Dec 2010
Location: T.Dot
Posts: 493
Re: The Official Math/Physics/Whatever Homework questions thread

Quote:
 Originally Posted by coffee_monster If the demand were 11, but AHOLE only had 10 units, how many would they sell? How much profit or revenue would they get. If the demand were 12? Now, if AHOLE had 11 units, ...
im lost, sorry i dont follow?

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