Gretchen has eight socks, two of each color: magenta, cyan, black, and white. She randomly draws four socks. What is the probability that sh

Gretchen has eight socks, two of each color: magenta, cyan, black, and white.

She randomly draws four socks.

What is the probability that she has exactly one pair of socks with the same color?

\(\text{Let ${\color{magenta}{m}} = {\color{magenta}{magenta}}$ } \\ \text{Let ${\color{cyan}{c}} = {\color{cyan}{cyan}}$ } \\ \text{Let ${\color{black}{b}} = {\color{black}{black}}$ } \\ \text{Let ${\color{grey}{w}} = {\color{grey}{white}}$ } \)

The Set is\(\{ {\color{magenta}{m_1}},{\color{magenta}{m_2}}, {\color{cyan}{c_1}},{\color{cyan}{c_2}}, {\color{black}{b_1}},{\color{black}{b_2}}, {\color{grey}{w_1}},{\color{grey}{w_2}} \} \)

The number of all the possibilities is \(^8C_4=\dbinom{8}{4} = \mathbf{70 }\)

\(\begin{array}{|rcll|} \hline && \dfrac{ \dbinom{ {\color{magenta}{2} } }{ 2 }\times \left[ \dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{0} +\dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{0}\dbinom{\color{grey}{2}}{1} +\dbinom{{\color{cyan}{2}}}{0}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{1} \right] \\ + \dbinom{ {\color{cyan}{2} } }{ 2 }\times \left[ \dbinom{{\color{magenta}{2}}}{1}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{0} +\dbinom{{\color{magenta}{2}}}{1}\dbinom{{\color{black}{2}}}{0}\dbinom{\color{grey}{2}}{1} +\dbinom{{\color{magenta}{2}}}{0}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{1} \right] \\ + \dbinom{ {\color{black}{2} } }{ 2 }\times \left[ \dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{magenta}{2}}}{1}\dbinom{\color{grey}{2}}{0} +\dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{magenta}{2}}}{0}\dbinom{\color{grey}{2}}{1} +\dbinom{{\color{cyan}{2}}}{0}\dbinom{{\color{magenta}{2}}}{1}\dbinom{\color{grey}{2}}{1} \right] \\ + \dbinom{ {\color{grey}{2} } }{ 2 }\times \left[ \dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{magenta}{2}}{0} +\dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{0}\dbinom{\color{magenta}{2}}{1} +\dbinom{{\color{cyan}{2}}}{0}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{magenta}{2}}{1} \right] } {70} \\\\ &=& \dfrac{ 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] \\ + 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] \\ + 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] \\ + 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] } {70} \\\\ &=& \dfrac{ ( 4+4+4 ) + (4+4+4) + (4+4+4) + (4+4+4) } {70} \\ &=& \dfrac{ 12 + 12 + 12 + 12 } {70} \\ &=& \dfrac{ 4\cdot 12 } {70} \\\\ &=& \dfrac{ 2\cdot 12 } {35} \\\\ &\mathbf{=}&\mathbf{ \dfrac{ 24 } {35} } \\ \hline \end{array}\)

The probability that she has exactly one pair of socks with the same color is \(\mathbf{ \dfrac{ 24 } {35} }\)

There are 4 possible pairs Gretchen can pick, and (6C2)−3=12 ways for her to pick socks of two other colors. There are (8C4)=70 total ways Gretchan can pick socks, and thus there is a 4⋅12/70 =24/35 probability Gretchen picks exactly 1 pair.

This number might seen large, but considering there is a 2^4/ 70 =8/35 chance of picking no pairs, and a (4C2)/70 =3/35 chance of two pairs, it adds up (to 1).

https://math.stackexchange.com/questions/2660495/probability-of-choosing-exactly-1-pair-out-of-4