Re: discrete math with balls.
- From: Dave Seaman <dseaman@xxxxxxxxxxxx>
- Date: Fri, 1 Dec 2006 14:54:57 +0000 (UTC)
On Fri, 1 Dec 2006 05:52:33 -0800, William Elliot wrote:
On Fri, 1 Dec 2006, mina_world wrote:
There are 100 black balls and 100 white balls in bag(or urn).
I take out 3 balls from bag. then I put it into a wastebasket.
1)
If I took out 3 black balls and 0 white ball,
I put 1 black ball and 0 white ball into bag.
Doesn't change parity of black or white.
And decreases the total number of balls, but does not change the number
of white.
2)
If I took out 2 black balls and 1 white ball,
I put 1 black ball and 1 white ball into bag.
Changes parity of black, doesn't change parity of white.
And decreases the total number of balls, but does not change the number
of white.
3)
If I took out 1 black ball and 2 white balls,
I put 0 black ball and 2 white balls into bag.
Changes parity of black, doesn't change parity of white.
And decreases the total number of balls, but does not change the number
of white.
4)
If I took out 0 black balls and 3 white balls,
I put 1 black ball and 1 white ball into bag.
Changes parity of black, doesn't change parity of white.
And decreases the total number of balls, and decreases the number of
white balls by 2.
What happens if the first time you take out 0 black and 3 white?
How can you put 1 black ball back?
The word "back" appears nowhere in the problem statement. The balls that
were removed were discarded, and therefore there must be a supply of
fresh balls.
This is a rule.
and iteration...
Find the (smallest) number of balls in bag after final step.
What final step? How are you assured this will stop?
The number of balls decreases with each step. When fewer than three
balls are left, the process must stop.
You aren't. Now prove with probablity zero, that the
process with go on without end.
The process must terminate after at most 98 steps. Therefore, the
probability of an unending process is zero.
answer : two white balls.
Answer: with probability 1, the sequence of events will terminate
in finite time by the bag having only two balls.
That the balls are both white is answer to a different question.
The number of white balls is always even, can never decrease by more than
2 at any step, and cannot decrease at all unless at least 3 white balls
are present.
I want to know the reason.
so, I need your explanation.
There will always be an even number of whites.
Show that they can never be none.
(Not possible if rule 3 is put one black back, 0 white back)
Show also that all the blacks can be removed.
(Not possible if rule 3 is put one black back, 2 white back)
Hard problem. What's the expected number of 'tooks' for termination.
Fewer than 98.
--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.
- References:
- discrete math with balls.
- From: mina_world
- Re: discrete math with balls.
- From: William Elliot
- discrete math with balls.
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