There are 8 white, 4 black and 2 orange balls. Random variable X denotes the winning amount. The values of X corresponding to otcomes of the event are: \(\displaystyle{W}{W}=-{2},{W}{O}=-{1},{O}{O}={0},{W}{B}={1},{B}{O}={2},{B}{B}={4}\)

Total number of choosing 2 balls from the urn containing 8+4+2 balls is 14C2. Number of choosing two white balls is 8C2 as they are only 8 white balls.

\(\displaystyle{P}{\left({x}=-{2}\right)}={\frac{{{n}_{{{W}{W}}}}}{{{n}_{{to{t}{a}{l}}}}}}={\frac{{{\frac{{{8}}}{{{2}}}}}}{{{\frac{{{14}}}{{{2}}}}}}}={\frac{{{28}}}{{{91}}}} \)

Number of choosing one white and one orange ball is \(\displaystyle{\left({1}{w}{h}{i}{t}{e}\right)}{\left({1}{\quad\text{or}\quad}{a}{n}ge\right)}\) as both are independent events.

\(\displaystyle{P}{\left({X}=-{1}\right)}={\frac{{{n}_{{{W}{O}}}}}{{{n}_{{to{t}{a}{l}}}}}}={\frac{{{\frac{{{8}}}{{{1}}}}{\frac{{{2}}}{{{1}}}}}}{{{\frac{{{14}}}{{{2}}}}}}}={\frac{{{16}}}{{{91}}}}\)

\(\displaystyle{P}{\left({X}={1}\right)}={\frac{{{n}_{{{W}{B}}}}}{{{n}_{{to{t}{a}{l}}}}}}={\frac{{{\frac{{{8}}}{{{1}}}}{\frac{{{4}}}{{{1}}}}}}{{{\frac{{{14}}}{{{2}}}}}}}={\frac{{{32}}}{{{91}}}}\)

\(\displaystyle{P}{\left({X}={0}\right)}={\frac{{{n}_{{{O}{O}}}}}{{{n}_{{to{t}{a}{l}}}}}}={\frac{{{\frac{{{2}}}{{{2}}}}}}{{{\frac{{{14}}}{{{2}}}}}}}={\frac{{{1}}}{{{91}}}}\)

\(\displaystyle{P}{\left({X}={2}\right)}={\frac{{{n}_{{{O}{B}}}}}{{{n}_{{to{t}{a}{l}}}}}}={\frac{{{\frac{{{2}}}{{{1}}}}{\frac{{{4}}}{{{1}}}}}}{{{\frac{{{14}}}{{{2}}}}}}}={\frac{{{8}}}{{{91}}}}\)

\(\displaystyle{P}{\left({X}={4}\right)}={\frac{{{n}_{{{B}{B}}}}}{{{n}_{{to{t}{a}{l}}}}}}={\frac{{{\frac{{{4}}}{{{2}}}}}}{{{\frac{{{14}}}{{{2}}}}}}}={\frac{{{6}}}{{{91}}}}\)