Question:

The probability that a randomly chosen 5-digit number formed from the digits 1, 2, 3, 4, 5 (without repetition) is divisible by 4, given that the number is even, is calculated as follows: First, determine the total number of even 5-digit numbers. A 5-digit number formed from the digits {1, 2, 3, 4, 5} without repetition is even if its last digit is 2 or 4. * **Case 1: The last digit is 2.** The remaining 4 digits (1, 3, 4, 5) can be arranged in the first four positions in $4!$ ways. Number of such numbers = $4! = 24$. * **Case 2: The last digit is 4.** The remaining 4 digits (1, 2, 3, 5) can be arranged in the first four positions in $4!$ ways. Number of such numbers = $4! = 24$. The total number of even 5-digit numbers is $24 + 24 = 48$. Next, determine the number of 5-digit numbers that are divisible by 4. A number is divisible by 4 if the number formed by its last two digits is divisible by 4. The digits must be distinct and chosen from {1, 2, 3, 4, 5}. The possible two-digit numbers formed from these digits that are divisible by 4 are: * Ending in 2: 12, 32, 52. * Ending in 4: 24. (Numbers ending in 1, 3, or 5 cannot be divisible by 4 and be divisible by 4). There are 4 such pairs for the last two digits: (1,2), (3,2), (5,2), (2,4). For each of these 4 pairs, the remaining 3 digits can be arranged in the first three positions in $3!$ ways. Number of 5-digit numbers divisible by 4 = $4 \times 3! = 4 \times 6 = 24$. Finally, calculate the conditional probability. Let $A$ be the event that the number is divisible by 4. Let $B$ be the event that the number is even. We want to find $P(A|B) = \frac{\text{Number of numbers in } A \cap B}{\text{Number of numbers in } B}$. Since any number divisible by 4 is also even, the event $A$ is a subset of event $B$ ($A \subset B$). Therefore, $A \cap B = A$. The number of favorable outcomes (numbers divisible by 4 and even) is the number of numbers divisible by 4, which is 24. The total number of outcomes in the reduced sample space (given that the number is even) is 48. The probability is: $$P(\text{divisible by 4 | even}) = \frac{24}{48} = \frac{1}{2}$$