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Old 10-24-2007, 08:28 PM
theblackkeys theblackkeys is offline
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Default On Girls Wearing Sweaters: A Pretty Problem with a Pretty Solution

Department of Mathematics & Statistics
CSU, Long Beach
Dr. Robert Mena
&
Dr. William Murray
Speaking on:
On Girls Wearing Sweaters: A Pretty Problem with a Pretty Solution

Friday October 26, 2007 12:00noon-1:00PM FO3-200A

Abstract:
Once upon a time, in the icy village of St. Yves lived 7 girls: Alison, Brenda, Chloe, Daphne, Emma, Fiona and Gabrielle. Each girl had 7 sweaters, one in each of seven colors: Orange, Purple, Rose, Scarlet, Turquoise, Wintergreen and Yellow.

Every Saturday when they go to the movies, each of the inseparable friends will wear one sweater. They have a ritual. The following Saturday to a given one, all of the girls will wear the same color sweater as the previous week, except for the follower (who is chosen at random for that week) who will wear a sweater of the same color as the leader (a different girl also chosen at random each week).

At the first movie each season the girls wear sweaters of different colors. Two questions:
1 In the long run what will occur?
2 How many weeks should one expect it will take before that happens?

A little reflection will lead to the answer of question 1. (For example, after the first week, the first follower’s original color is never seen again.) As it turns out, question 2 has a nice answer but to prove it in the general case of girls is slightly problematic. The talk will exhibit the outline of the general proof and it should be accessible to anyone who has a finished a course in matrix theory and has some basic understanding of probability (247 and 380 in CSULB terms).

/////////////
Unfortunately I'm not sure if I can go to this presentation, but was wondering if you guys wanted to work on this proof. Is this an interesting problem?
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Old 10-24-2007, 09:08 PM
drzen drzen is offline
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Default Re: On Girls Wearing Sweaters: A Pretty Problem with a Pretty Solution

They will all end up in the same colour sweater. I've no idea how long it will take.

It goes something like this though. I'm using letters for the colours and conventionally a is the leader first time, g the follower first time:

abcdefg

abcdefa

abbdefa

abbddfa

dbbddfa

and so on.
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Old 10-24-2007, 09:54 PM
mbillie1 mbillie1 is offline
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Default Re: On Girls Wearing Sweaters: A Pretty Problem with a Pretty Solution

girls who wear sweaters are probably prudes... the solution is to date skankier chicks
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Old 10-25-2007, 10:55 PM
MaxWeiss MaxWeiss is offline
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Default Re: On Girls Wearing Sweaters: A Pretty Problem with a Pretty Solution

Well each iteration will have 1 out of 7 wearing the same as one of the other 6 colors. So after the first one, there will be a dominant color and a non-existent color. The dominant one has the greatest chance of being the final color, although another color could overtake it. My random guess is that it will take about 3-5 months for the final color to appear on all seven. I don't have a matrix or formula worked out, that's just my intuitive guess.

Edit: I am assuming that each week follows the pattern, even though in the OP it said following a given week, which could start at any time.
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Old 10-28-2007, 12:40 AM
pzhon pzhon is offline
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Default Re: On Girls Wearing Sweaters: A Pretty Problem with a Pretty Solution

This is an interesting problem, although I think I've seen it before, and remembered the general form of the answer. It might be that I saw this from Robert Mena, since I've gone to a few of his talks and had lunch with him, but I thought I saw it elsewhere.

Spoiler in white: <font color="white">The expected length of time before all sweaters are the same color is 36 = (7-1)^2. This pattern seems to hold for more general values of 7, and that is what I remember, but I don't see a complete proof yet.

My computation for 7 colors:
Let X be a random variable taking the value n if it takes n weeks for all colors to match. We want to determine E(X)
Let X_c be a random variable taking the value n if color c is the winner after n weeks, and 0 otherwise. X = Sum X_c, so by the linearity of expected value, E(X) = Sum E(X_c). By symmetry, this is 7 E(X_1).

Suppose we start with i sweaters of color 1, and j=7-i sweaters of color 2. Let T(i,j) be the expected value of a random varuable which is n if color 1 wins after n weeks, and 0 if color 2 wins. T(0,7) = T(7,0)= 0. What we would like to compute is E(X_1)=T(1,6), since we can ignore the differences between the other colors.

T(i,j) =
(no change) (1-ij/21) ((prob win) i/7 + (recurse) T(i,j))
+ (gain) (ij/42) ((prob win) (i+1)/7 + (recurse) T(i+1,j-1))
+ (loss) (ij/42) ((prob win) (i-i)/7 + (recurse) T(i-1,j+1))

The solution to this system of equations is

T(0,7) = 0
T(1,6) = 36/7
T(2,5) = 65/7
T(3,4) = 428/35
T(4,3) = 957/70
T(5,2) = 459/35
T(6,1) = 669/70
T(7,0) = 0

So, the expected amount of time before all sweaters are the same color is 7 * 36/7 = 36.

I think there is supposed to be a relatively simple proof by a different method, although it sounds like the talk was probably along these lines with a linear algebra shortcut to compute only T(1,6).
</font>
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