Mixed in a drawer are 2 blue​ socks, 4 white​ socks, and 6 gray socks. You pull out two​ socks, one at a​ time, without looking. Find the probability of getting 2 socks of the same color.

Question

Thank you for helping!

Answer 1

One way to approach this problem is to think about what all the possible outcomes are, and then consider which ones we need to solve the problem.

Starting with the first sock, we have 3 possible outcomes: blue, white, or gray. If we were wondering what the probability was of just drawing one blue sock, we could get that by dividing the number of blue socks in the drawer by the total number of socks ( 2 / 12 ).

For the second sock, we also have three outcomes: blue, white, or gray. Keep in mind we are drawing without replacement, so there is one less sock in the drawer at this point. So since we have 3 options for the first sock and 3 options for the second sock, the total number of combinations is 3 * 3 = 9. Here are all the possible combinations: (blue, blue), (blue, white), (blue, gray), (white, blue), (white, white), (white, gray), (gray, blue), (gray, white), (gray, gray).

As you can see, there are only 3 outcomes out of the 9 possible outcomes where both socks are the same color. Each of these outcomes are independent of each other. This means we can add up the individual probabilities of each of these 3 specific outcomes (2 blue socks, 2 white socks, or 2 gray socks) to get the total probability (2 same-color socks).

To get the probability of getting multiple socks (first sock, second sock), multiply their individual probabilities together.

For example, P(blue sock first & white sock second) = P(blue sock first) * P(white sock second) = 2/12 * 4/11 = 8/132 = 2/33 (remember there is one less sock in the drawer after you draw the first one).

So the probability of 2 socks of the same color is, in equation form:

P(2 socks of same color) = P(blue sock first) * P(blue sock second) + P(white sock first) * P(white sock second) + P(gray sock first) * P(gray sock second).

Hopefully this is enough to help you finish the problem. Please let me know if you have any more questions!