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: \[ \boxed{\frac{2}{9}} \]