Just 2 random process!

Probability Level 3

There are m m white and m m black balls in a big bag (where m > 0 m>0 ). Now randomly choose m m balls to move into another big bag.

From 2 bags, a ball is randomly chosen from each bag. Find the probability that both balls have the same color.

m + 1 2 m + 1 \frac{m+1}{2m+1} m − 1 2 m + 1 \frac{m-1}{2m+1} m − 1 2 m − 1 \frac{m-1}{2m-1} m + 1 2 m − 1 \frac{m+1}{2m-1}

Raymond Chan by Raymond Chan , 20, USA

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1 solution

Vaibhav Thakkar
Jul 26, 2018

T h e Q u e s t i o n i s e q u i v a l e n t o f t a k i n g 2 b a l l s a t r a n d o m f r o m t h e b a g a n d f i n d i n g t h e P r o b a b i l t y t h a t b o t h o f t h e m a r e o f s a m e c o l o u r . L e t E b e t h e e v e n t t h a t t w o b a l l a r e t a k e n o u t a n d r a n d o m f r o m t h e b a g a n d t h e y a r e o f s a m e c o l o u r . ⇒ P ( E ) = m C 2 + m C 2 2 m C 2 = m − 1 2 m − 1 \\The\ Question\ is\ equivalent\ of\ taking\ 2\ balls\ at\ random\ from\ the\ bag \ and\\ finding\ the\ Probabilty\ that\ both\ of\ them\ are\ of\ same\ colour. \\\\ Let\ E\ be\ the\ event\ that\ two\ ball\ are\ taken\ out\ and\ random\ from\ the\ bag\ and\\ they\ are\ of\ same\ colour. \\\\ \Rightarrow P(E) = \frac{\ ^{m}C_{2}\ +\ ^{m}C_{2}}{^{2m}C_{2}}\\\\ =\ \frac{m-1}{2m-1}

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