1. **Problem Statement:** An experiment consists of asking 3 persons at random if they wash their clothes using detergent. (a) List the elements of a sample space $S$ using the letter $Y$ for ‘yes’ and $N$ for ‘no’. (b) List the elements of $S$ corresponding to event $E$ that at least 2 of the persons use brand detergent. 2. **Formula and Explanation:** - The sample space $S$ consists of all possible outcomes of the 3 persons answering either $Y$ or $N$. Since each person has 2 possible answers, the total number of outcomes is $2^3 = 8$. - Event $E$ is the subset of $S$ where at least 2 persons answer $Y$. This means either exactly 2 persons say $Y$ or all 3 say $Y$. 3. **Step-by-step Solution:** (a) Sample space $S$: List all combinations of $Y$ and $N$ for 3 persons: $$S = \{YYY, YYN, YNY, NYY, YNN, NYN, NNY, NNN\}$$ (b) Event $E$ (at least 2 $Y$): - Exactly 2 $Y$s: $YYN, YNY, NYY$ - Exactly 3 $Y$s: $YYY$ So, $$E = \{YYY, YYN, YNY, NYY\}$$ 4. **Explanation:** - The sample space lists every possible combination of answers. - Event $E$ includes all outcomes where the number of $Y$s is 2 or 3. - This is a common approach in probability to define events based on counts of outcomes. **Final answers:** (a) $S = \{YYY, YYN, YNY, NYY, YNN, NYN, NNY, NNN\}$ (b) $E = \{YYY, YYN, YNY, NYY\}$