My drawer contains 4 blue socks, 7 red socks, and 3 yellow socks. If I randomly pull 2 socks at the same time, what is the probability that the socks are the same color?
I know that the probability that the first sock is blue is $\frac4{14}=\frac27
$. But I do not know how to calculate the probability that the first two socks are blue.
Best Answer
As Adriano pointed, we gonna split this problem in 3 sub-problems. What's the probability of drawing a 2 blue socks in the first 2 drawing. It's:
$$\frac{4}{14} \times \frac{3}{13} = \frac{12}{182}$$
Now what's the probability of drawing 2 red socks? It's:
$$\frac{7}{14} \times \frac{6}{13} = \frac{42}{182}$$
And the final sub-problem, what's the probability of drawing 2 yellow socks? It's:
$$\frac{3}{14} \times \frac{2}{13} = \frac{6}{182}$$
Now we add up this 3 fractions and we end up with:
$$\frac{12}{182} + \frac{42}{182} + \frac{6}{182} = \frac{60}{182} \approx 32.97 \%$$