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#1
10-11-2005, 06:33 PM
 allintuit Member Join Date: Apr 2005 Posts: 67
Extremely Difficult Probability Problem

Suppose you are truthfully told that ten marbles were inserted into a box, all of them identical except that their colors were determined by the toss of an unbiased coin. When heads came up, a white marble was inserted, and when tails came up, a black one. You reach into the box, draw out a marble, inspect its color, then return it to the box. You shake the box to mix the marbles randomly, and then reach in and again select a marble at random. If you inspect ten marbles in succession in this manner and all turn out to be white, what is the probability to the nearest whole percent that all ten marbles in the box are white?

This problem has stumped me.
#2
10-11-2005, 06:47 PM
 KJL Senior Member Join Date: Jun 2005 Posts: 135
Re: Extremely Difficult Probability Problem

Edit: I am dumb.
#3
10-11-2005, 06:53 PM
 allintuit Member Join Date: Apr 2005 Posts: 67
Re: Extremely Difficult Probability Problem

(-_-).
#4
10-11-2005, 07:02 PM
 David Sklansky Senior Member Join Date: Aug 2002 Posts: 241
Re: Extremely Difficult Probability Problem

You must figure out ten results:

1. The chances that there is only one white marble in there and that you picked it ten times (10/1024 x 1/10,000,000,000).

2. The chances that there are two white marbles in there and you picked one of them ten times (45/1024 x 1024/10,000,000,000)

3. The chances that there are three white marbles in there and you picked one of them ten times (Ten choose three, divided by two to the tenth power, times the quantity three to the tenth power, divided by ten to the tenth power).

Same procedure up to nine

10. The chances that there are ten white balls in there times the chances you picked ten white balls is simply 1/1024 times 1.

Final answer is 1/1024 (The number 10 result) divided by the sum of all the results.
#5
10-11-2005, 07:07 PM
 KJL Senior Member Join Date: Jun 2005 Posts: 135
Re: Extremely Difficult Probability Problem

#6
10-11-2005, 07:29 PM
 allintuit Member Join Date: Apr 2005 Posts: 67
Re: Extremely Difficult Probability Problem

[ QUOTE ]
You must figure out ten results:

1. The chances that there is only one white marble in there and that you picked it ten times (10/1024 x 1/10,000,000,000).

2. The chances that there are two white marbles in there and you picked one of them ten times (45/1024 x 1024/10,000,000,000)

3. The chances that there are three white marbles in there and you picked one of them ten times (Ten choose three, divided by two to the tenth power, times the quantity three to the tenth power, divided by ten to the tenth power).

Same procedure up to nine

10. The chances that there are ten white balls in there times the chances you picked ten white balls is simply 1/1024 times 1.

Final answer is 1/1024 (The number 10 result) divided by the sum of all the results.

[/ QUOTE ]

David,

Good job there, I never thought of doing that. Thanks!
#7
10-11-2005, 08:46 PM
 Guest Posts: n/a
Re: Extremely Difficult Probability Problem

you can get the same result by thinking of the question as being "what is the probability that a fair coin comes up the same all 10 times it is flipped" (i.e. all the marbles are the same color) which is (1/2)^10 or 1/1024
#8
10-12-2005, 01:56 AM
 kelvin474 Member Join Date: Jun 2003 Location: Inside I-495, 1.5 hours to FOXWOODS Posts: 96
Re: Extremely Difficult Probability Problem

I did a calculation which appears to be the same as what David said and I got something like 7.04%.

I then coded it up in MATLAB and ran a zillion trials and got 6.99% empirically. Standard error was .048% so it appears reasonable.

The answer is 7%. The most likely was that 8 of 10 are white.
#9
10-12-2005, 11:57 AM
 alThor Junior Member Join Date: Mar 2004 Posts: 6
Re: Extremely Difficult Probability Problem

[ QUOTE ]
I did a calculation which appears to be the same as what David said and I got something like 7.04%.

[/ QUOTE ]

Agree. (It's closer to 7.02%.)

alThor
#10
10-12-2005, 12:55 PM
 mosdef Senior Member Join Date: Jan 2005 Location: Toronto Posts: 168
Re: Extremely Difficult Probability Problem

[ QUOTE ]
you can get the same result by thinking of the question as being "what is the probability that a fair coin comes up the same all 10 times it is flipped" (i.e. all the marbles are the same color) which is (1/2)^10 or 1/1024

[/ QUOTE ]

no, this is the probability before you do any of the removal of marbles and checking that they are white. after you've sampled 10 times and seen 10 whites, the probability is different.

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