Simple probability

[Mickey]

How would you divide 50 black and 50 white marbles into two piles so
that the probability of picking a white marble as follows is
maximized: we first pick one of the piles, then we pick a marble in
that pile?

[siddharth agrawal]

One pile with 1 white marble and other pile with 50 black and 49 white marbles..

Regards,

Siddharth

[Mickey]

That is the correct answer. How did you conclude it? Can it be shown
mathematically that this is optimal?

On May 18, 4:25 pm, siddharth agrawal <4u.siddha...@gmail.com> wrote:
> One pile with 1 white marble and other pile with 50 black and 49 white
> marbles..
>
> Regards,
> Siddharth
>

[SUDHEER DURUSOJU]

Probablity to pick a pile is 1/2 ( Need to pick 1 out of 2 ).

Probability to pick white marble is 1/2 * 1/x where x stands for number of white marbles in that pile.

1/2 * 1/x is optimal when x=1.

-Sudheer

[siddharth agrawal]

Hi jyoti,

m not sure about the mathematical explanation to prove it optimal..it just clicked..

but I don't think that the explanation provided by sudheer is correct..( 1/x part).

If 1st pile contains 1 white marble then according to ur explanation x=1,

and hence the probability to pick up white marble from 2nd bag is 1/49 (which is not correct).

It is actually 49/(49+50)=49/99

Here 1/2 'll be common in all the cases(probability to pick one of the bags)..so we need to maximise the probability 

of picking white marbles. if first pile contains n number of marbles then probability for first bag is n/n and for second bag is (50-n)/(100-n)[Considering that all the blacks are in second bag]. I jus tried to maximise that value.(50-n)/(100-n) can also be written as {1 - (50/(100-n))}.Maximum value will be at n=1(n can not be 0 coz for first bag probability is n/n). This is all i cud think of.

Please tell me if i'm wrong and provide the mathematical solution if possible.

Regards,

Siddharth

[SUDHEER DURUSOJU]

Siddharth,

I agree with you. I was thinking in similar lines but did missed lot of info in my previous mail.

-Sudheer

[Mickey]

I had taken the question from here:
http://gurmeetsingh.wordpress.com/2008/09/16/puzzle-divide-100-marbles-into-two-piles/
The solution is also listed here.

Here is how I solved it:
let's say p marbles in first pile and among them w is white. so the
second pile has 50-w white marbles and 100-p marbles in total.
The probability of picking up a white marble is given by:
1/2 * (w/p + (50-w)/(100-p))
with the constraints:
1. p >= w (you can't have, say, 10 white marbles in the pile and less
count of total marbles)
2. w <= 50 (because there aren't more white marbles)
3. p < w + 50 (I hope this is also clear)
4. p and w are positive integers.

We want to maximize the above expression keeping in mind the
constraints.
With little inspection we figure out that minimizing p maximizes the
above expression. And the smallest integral p that is possible is 1.


Regards,
Jyoti

On May 19, 8:10 pm, SUDHEER DURUSOJU <sdurus...@gmail.com> wrote:
> Siddharth,
>
> I agree with you. I was thinking in similar lines but did missed lot of info
> in my previous mail.
>
> -Sudheer
>
> On Wed, May 19, 2010 at 10:54 AM, siddharth agrawal

> <4u.siddha...@gmail.com>wrote:

>
> > Hi jyoti,
>
> > m not sure about the mathematical explanation to prove it optimal..it just
> > clicked..
> > but I don't think that the explanation provided by sudheer is correct..(
> > 1/x part).
>
> > If 1st pile contains 1 white marble then according to ur explanation x=1,
> > and hence the probability to pick up white marble from 2nd bag is 1/49
> > (which is not correct).
> > It is actually 49/(49+50)=49/99
>
> > Here 1/2 'll be common in all the cases(probability to pick one of the
> > bags)..so we need to maximise the probability
> > of picking white marbles. if first pile contains n number of marbles then
> > probability for first bag is n/n and for second bag is
> > (50-n)/(100-n)[Considering that all the blacks are in second bag]. I jus
> > tried to maximise that value.(50-n)/(100-n) can also be written as {1 -
> > (50/(100-n))}.Maximum value will be at n=1(n can not be 0 coz for first bag
> > probability is n/n). This is all i cud think of.
>
> > Please tell me if i'm wrong and provide the mathematical solution if
> > possible.
>
> > Regards,
> > Siddharth
>

> > On Wed, May 19, 2010 at 6:56 PM, SUDHEER DURUSOJU <sdurus...@gmail.com>wrote:
>
> >> Probablity to pick a pile is 1/2 ( Need to pick 1 out of 2 ).
>
> >> Probability to pick white marble is 1/2 * 1/x where x stands for number of
> >> white marbles in that pile.
>
> >> 1/2 * 1/x is optimal when x=1.
>
> >> -Sudheer
>