Question

Suppose a drawer contains six white socks, three brown socks, and seven black socks. We draw one sock from the drawer and it is equally likely that any one of the socks is drawn. We reach into the drawer looking to pull out four socks. What is the probability that we get at least two socks of the same color?

Answer 1

Answer:

The probability is $1$.

Step-by-step explanation:

We have a drawer that contains :

• Six white socks
• Three brown socks
• Seven black socks

Let's define the following events :

W : '' The sock we got was white ''

Br : '' The sock we got was brown ''

Bl : '' The sock we got was black ''

We can calculate the following probabilities by counting :

$P(W)=\frac{6}{16}$

Because there are 6 white socks out of 16 socks in the drawer

Using the same reasoning :

$P(Br)=\frac{3}{16}$

$P(Bl)=\frac{7}{16}$

Now to calculate the probability of getting at least two socks of the same color we need to think about the following :

Getting at least two socks of the same color means getting two, three or four (except the brown socks because we only have three brown socks) of the same color.

Again we can calculate the probability by counting :

$\frac{FavorableCases}{TotalCases}$

But giving that we pull out four socks, we always will get at least two socks of the same color.

In the most desfavorable case we will get one white sock, one brown sock and one black sock. And in the last extraction we will complete it with a white, brown or black sock (obtaining two socks of the same color).

Finally, all extractions of four socks will have at least two of the same color.

The probability is $1$.