There are 5 red socks, 2 white socks, and 3 blue socks in a basket. What is the probability of selecting one red sock from the basket, then selecting a second red sock from the basket without replacing the first?

1 answer

To find the probability of selecting two red socks in succession without replacement, we first determine the total number of socks and the number of red socks.

Initially, the basket contains:

  • 5 red socks
  • 2 white socks
  • 3 blue socks

The total number of socks in the basket is: \[ 5 + 2 + 3 = 10 \text{ socks} \]

Step 1: Probability of selecting the first red sock

The probability of selecting a red sock first is given by the ratio of the number of red socks to the total number of socks: \[ P(\text{first red sock}) = \frac{5}{10} = \frac{1}{2} \]

Step 2: Probability of selecting the second red sock

After selecting the first red sock, there are now:

  • 4 red socks remaining (since one red sock has been taken)
  • 2 white socks
  • 3 blue socks

Thus, the new total number of socks in the basket becomes: \[ 4 + 2 + 3 = 9 \text{ socks} \]

Now, the probability of selecting a second red sock, given that the first sock selected was red, is: \[ P(\text{second red sock} \mid \text{first red sock}) = \frac{4}{9} \]

Step 3: Combined probability

To find the overall probability of selecting two red socks in succession, we multiply the probabilities of both events: \[ P(\text{two red socks}) = P(\text{first red sock}) \times P(\text{second red sock} \mid \text{first red sock}) = \frac{1}{2} \times \frac{4}{9} = \frac{4}{18} = \frac{2}{9} \]

Thus, the probability of selecting two red socks in succession without replacement is: \[ \boxed{\frac{2}{9}} \]