To solve the problem, we need to find the probability that a randomly selected integer from the set \( U \) is not a multiple of 4. Let's break this down step-by-step. ### Step 1: Define the Universal Set \( U \) The universal set \( U \) is defined as: \[ U = \{ x : x \text{ is an integer and } 1 \leq x \leq 20 \} \] This means \( U \) contains all integers from 1 to 20. Therefore, the total number of elements in \( U \) is: \[ |U| = 20 \] ### Step 2: Define the Set \( E2 \) The set \( E2 \) is defined as: \[ E2 = \{ x : x \text{ is a multiple of 4} \} \] To find the elements of \( E2 \) within the range of \( U \), we list the multiples of 4 from 1 to 20: - The multiples of 4 are: 4, 8, 12, 16, and 20. Thus, the elements of \( E2 \) are: \[ E2 = \{ 4, 8, 12, 16, 20 \} \] The number of elements in \( E2 \) is: \[ |E2| = 5 \] ### Step 3: Find the Complement of \( E2 \) To find the probability that a randomly selected integer from \( U \) is **not** in \( E2 \), we need to determine the complement of \( E2 \), which consists of all integers in \( U \) that are not multiples of 4. The total number of integers in \( U \) is 20, and the number of integers that are multiples of 4 (i.e., the size of \( E2 \)) is 5. Therefore, the number of integers that are **not** multiples of 4 is: \[ |U| - |E2| = 20 - 5 = 15 \] ### Step 4: Calculate the Probability The probability \( P \) that a randomly selected integer from \( U \) is not in \( E2 \) is given by the ratio of the number of favorable outcomes (not in \( E2 \)) to the total number of outcomes (in \( U \)): \[ P(\text{not in } E2) = \frac{|U| - |E2|}{|U|} = \frac{15}{20} \] This fraction can be simplified: \[ P(\text{not in } E2) = \frac{15 \div 5}{20 \div 5} = \frac{3}{4} \] ### Conclusion: Correct Option Thus, the correct answer is: **A. \( \frac{3}{4} \)** ### Step 5: Explanation of Other Options - **B. \( \frac{3}{10} \)**: This option is incorrect because it does not reflect the correct ratio of integers not in \( E2 \) to the total integers in \( U \). - **C. \( \frac{1}{4} \)**: This option represents the probability of selecting a multiple of 4, which is the opposite of what we are looking for. - **D. \( \frac{1}{20} \)**: This option is far too small and does not represent any meaningful probability in this context. ### Revision Summary - The universal set \( U \) contains integers from 1 to 20, totaling 20 elements. - The set \( E2 \) includes multiples of 4 within that range, totaling 5 elements. - The probability of selecting an integer not in \( E2 \) is calculated as \( \frac{15}{20} = \frac{3}{4} \). - The correct answer is option A: \( \frac{3}{4} \).