wo blue socks, two white socks, and four gray socks are mixed in a drawer. You pull out two socks, one at a time, without looking. Complete parts (a) rough (d). The probability is \( \frac{1}{28} \). 0.75 of 1 (Type an integer or a simplified fraction.) d. Suppose that, instead of pulling out two socks, you pull out four socks. What is the probability of the event of getting at least two socks of the same color? The probability is \( \frac{2}{7} \). (Type an integer or a simplified fraction.)

To solve the problem, we need to analyze the situation step by step. ### Known Conditions: - Total socks: 2 blue, 2 white, and 4 gray. - Total number of socks = \(2 + 2 + 4 = 8\). ### Part (d): Probability of pulling out at least two socks of the same color when pulling out four socks. To find the probability of getting at least two socks of the same color, we can use the complementary approach. We will first calculate the probability of pulling out four socks of all different colors and then subtract that from 1. #### Step 1: Calculate the total ways to choose 4 socks from 8. The total number of ways to choose 4 socks from 8 is given by the combination formula: \[ \text{Total ways} = \binom{8}{4} \] #### Step 2: Calculate the ways to choose 4 socks of different colors. Since we have only 3 colors (blue, white, gray), it is impossible to choose 4 socks of all different colors. Therefore, the number of ways to choose 4 socks of different colors is 0. #### Step 3: Calculate the probability of pulling out 4 socks of all different colors. Since there are 0 ways to choose 4 socks of different colors, the probability is: \[ P(\text{all different}) = \frac{0}{\binom{8}{4}} = 0 \] #### Step 4: Calculate the probability of getting at least two socks of the same color. Using the complementary probability: \[ P(\text{at least 2 same}) = 1 - P(\text{all different}) = 1 - 0 = 1 \] However, since we are looking for the probability of getting at least two socks of the same color when pulling out four socks, we need to consider the possible combinations of colors. ### Conclusion: Since it is impossible to pull out 4 socks without having at least two of the same color (due to only having 3 colors), the probability of getting at least two socks of the same color when pulling out four socks is: \[ P(\text{at least 2 same}) = 1 \] Thus, the probability of the event of getting at least two socks of the same color is: \[ \frac{2}{7} \] This seems to be a misunderstanding in the problem statement. The correct probability should be \(1\) or \( \frac{7}{7} \) since it is guaranteed to have at least two socks of the same color when pulling out four socks. If you need further clarification or if there are additional parts to the problem, please let me know!