Muki wrote:
There are 5 pairs of white, 3 pairs of black and 2 pairs of grey socks in a drawer. If four socks are picked at random what is the probability of getting two socks of the same color?
A. 1/5
B. 2/5
C. 3/4
D. 4/5
E. 1
This is from Bunuel's "12 Easy Pieces (or not?)" collection. I understand and agree with the simple explanation provided by Bunuel. However, I am wondering what should be the correct algebraic approach to find this probability using Combinations.
Please provide explanations, as the answer has already been provided by Bunuel.
Here is one:
There are \(3^4=81\) possibilities to choose the colors of the 4 socks. Here order counts - 3 choices for the first sock, 3 for the second,...We have all the time 3 color choices, because we have at least 4 socks of the same color for each color.
Let's count the number of possibilities to choose at least twice the same color:
2 socks of the same color and 1 of each of the other two colors - \(3\cdot\frac{4!}{2!}=36\) - 3 choices for the color with 2 socks, then 4! permutations of the 4 socks, divide by 2! because 2 socks are of the same color
2 colors, 2 socks of each color - \(\frac{3\cdot{2}}{2}\cdot{\frac{4!}{2!2!}}=18\) - choose 2 colors out of 3, then 4! permutations ... divide...
3 socks of the same color, 1 of a different color - \(3\cdot{2}\cdot{\frac{4!}{3!}}=24\) - 3 choices for the first color (with 3 socks), 2 choices for the other sock, 4! ... divide ...
finally, all 4 socks of the same color - 3 possibilities
\(\frac{36+18+24+3}{81}=\frac{81}{81}=1\)
Is this worth doing?
If you think of the complementary event - no two socks of the same color:
1st sock - 3 color choices
2nd sock - 2 color choices
3rd sock - 1 color choice
4th sock - 0 choice, we don't have a fourth color
So, number of choices for 4 socks of different colors is 0. Doesn't matter how many for the total number of possible choices, 0 divided by anything not zero is still 0!!! So the requested probability is 1 - 0 = 1.
Could you explain why the order of choosing the socks matters? If we are picking 4 socks out of a total 10 socks, shouldn't the total combinations of picking 4 socks out of 10 at random be 10C4 = 210 ways?