0. There are two black socks and one gray sock in a drawer. Without looking, Phil selects two socks. a. Find the sample space of the experiment b. What is the probability that Phil selects two black socks?

To solve this problem, we will follow the steps outlined: ### Step 1: Extract Known Conditions - There are 2 black socks (let's denote them as \( B_1 \) and \( B_2 \)). - There is 1 gray sock (denote it as \( G \)). - Phil selects 2 socks without looking. ### Step 2: Find the Sample Space The sample space consists of all possible combinations of selecting 2 socks from the total of 3 socks (2 black and 1 gray). The possible combinations are: 1. \( (B_1, B_2) \) - both black socks 2. \( (B_1, G) \) - one black sock and one gray sock 3. \( (B_2, G) \) - one black sock and one gray sock Thus, the sample space \( S \) is: \[ S = \{ (B_1, B_2), (B_1, G), (B_2, G) \} \] ### Step 3: Calculate the Probability of Selecting Two Black Socks To find the probability that Phil selects two black socks, we need to determine the number of favorable outcomes and the total number of outcomes. - **Favorable outcomes**: There is only 1 favorable outcome, which is \( (B_1, B_2) \). - **Total outcomes**: There are 3 outcomes in the sample space. The probability \( P \) of selecting two black socks is given by the formula: \[ P(\text{two black socks}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{3} \] ### Final Answers a. The sample space of the experiment is: \[ S = \{ (B_1, B_2), (B_1, G), (B_2, G) \} \] b. The probability that Phil selects two black socks is: \[ P(\text{two black socks}) = \frac{1}{3} \]