2D probability question..

in a jar we have 3 balls one red one white one black

3 people choose a ball and return it back.

x represents the number of different colors which the 3 people chose
y represents the number of white balls chosen.

p(x,y)
find p(2,0)

we have in total 27 possibilities
3 people chose 2 different colors(from the existing 3) and non of them was white

so each guy need to chose red or black out of 3 balls
$\displaystyle \binom{3}{1}+\binom{3}{1}$

why the answer is $\displaystyle \frac{2\binom{3}{2}}{27}$

Quote:

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Originally Posted by transgalactic

in a jar we have 3 balls one red one white one black

3 people choose a ball and return it back.

x represents the number of different colors which the 3 people chose
y represents the number of white balls chosen.

p(x,y)
find p(2,0)

we have in total 27 possibilities
3 people chose 2 different colors(from the existing 3) and non of them was white

so each guy need to chose red or black out of 3 balls
$\displaystyle \binom{3}{1}+\binom{3}{1}$

why the answer is $\displaystyle \frac{2\binom{3}{2}}{27}$

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Actually, the correct answer is $\displaystyle \frac{2}{9}$! (Giggle)

Note that $\displaystyle \binom{3}{2}= \binom{3}{1}= 3$ so the only difference between your answer and the given answer is the "27" in the denominator. That is there, of course, because there are a total of 27 possible outcomes:
$\displaystyle \frac{\binom{3}{1}+ \binom{3}{1}}{27}= \frac{2\binom{3}{2}}{27}= \frac{(2)(3)}{27}= \frac{2}{9}$.