A drawer of loose socks contains 2 red socks, 2 green socks, and 6 white socks. Which best describes how to determine the probability of pulling out a white sock, not replacing it, and pulling out another white sock?

Question

The probability that the first sock is white is (StartFraction 6 over 10 EndFraction) and that the second sock is white is (StartFraction 6 over 10 EndFraction), so the probability of choosing a pair of white socks is StartFraction 36 over 100 EndFraction = StartFraction 18 over 50 EndFraction.
The probability that the first sock is white is (StartFraction 1 over 10 EndFraction) and that the second sock is white is (StartFraction 1 over 10 EndFraction), so the probability of choosing a pair of white socks is StartFraction 1 over 100 EndFraction.
The probability that the first sock is white is (StartFraction 6 over 10 EndFraction) and that the second sock is white is (StartFraction 5 over 9 EndFraction), so the probability of choosing a pair of white socks is StartFraction 30 over 90 EndFraction = one-third.
The probability that the first sock is white is (StartFraction 1 over 10 EndFraction) and that the second sock is white is (StartFraction 1 over 9 EndFraction), so the probability of choosing a pair of white socks is StartFraction 1 over 90 EndFraction.

Answer 1

To determine the probability of pulling out a white sock, not replacing it, and then pulling out another white sock, we need to follow these steps:

1. Calculate the probability of drawing the first white sock:

   • There are 6 white socks out of a total of 10 socks (2 red + 2 green + 6 white).
   • The probability of pulling out the first white sock is: \[ P(\text{1st white}) = \frac{6}{10} \]

2. Calculate the probability of drawing the second white sock after removing the first one:

   • After pulling out the first white sock, there will be 5 white socks left and a total of 9 socks remaining (since one sock has already been removed).
   • The probability of pulling out the second white sock is: \[ P(\text{2nd white | 1st white}) = \frac{5}{9} \]

3. Calculate the combined probability of both events happening:

   • Since these two events are dependent (the outcome of the second event depends on the first), we multiply the probabilities: \[ P(\text{2 white socks}) = P(\text{1st white}) \times P(\text{2nd white | 1st white}) = \frac{6}{10} \times \frac{5}{9} = \frac{30}{90} = \frac{1}{3} \]

Thus, the correct description of how to determine the probability of pulling out two white socks in succession without replacement is:

The probability that the first sock is white is \(\frac{6}{10}\) and that the second sock is white is \(\frac{5}{9}\), so the probability of choosing a pair of white socks is \(\frac{30}{90}\) = one-third.