Socks from a Box - 3 matching socks
Let's say we have a box filled with socks: three pairs each of white, gray, and black socks (i.e. six socks of each of the three colors, for eighteen socks in total). We draw socks randomly from the box without replacement.
Let's say we draw three socks from the box. What is the probability that we draw the same color for all three socks?
$\endgroup$ 11 Answer
$\begingroup$Say you draw one sock. The probability of drawing any particular color, say black, is $\frac{6}{18}=\frac{1}{3}$. Then, there are $17$ socks remaining, of which only $5$ are black. The probability of drawing one of five black socks from $17$ total socks is $\frac{5}{17}$. Similarly, the probability of drawing one of four black socks from $16$ total socks is $\frac{4}{16}=\frac{1}{4}$.
Thus, the probability of drawing three black socks is $\frac{1}{3}\frac{5}{17}\frac{1}{4}=\frac{5}{204}\approx0.0245$.
Edit: As mentioned by @Ross in the comments, since the color choice does not matter as long as all three socks are the same color, the final answer is this times $3$, or $3\cdot\frac{1}{3}\frac{5}{17}\frac{1}{4}=\frac{5}{68}\approx0.0735$.
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