Question:

Step 1: 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 an even number. The even digits available are 2 and 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 $24 + 24 = 48$. Step 2: 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 available are {1, 2, 3, 4, 5} without repetition. The possible two-digit numbers formed from these digits that are divisible by 4 are: 12, 24, 32, 52. It is important to note that all these two-digit numbers end in an even digit (2 or 4). Therefore, any 5-digit number ending with one of these pairs will automatically be an even number. This means the count of numbers divisible by 4 is also the count of numbers that are both divisible by 4 and even. Case 1: The last two digits are 12. The remaining 3 digits (3, 4, 5) can be arranged in the first three positions in $3!$ ways. Number of such numbers = $3! = 6$. Case 2: The last two digits are 24. The remaining 3 digits (1, 3, 5) can be arranged in the first three positions in $3!$ ways. Number of such numbers = $3! = 6$. Case 3: The last two digits are 32. The remaining 3 digits (1, 4, 5) can be arranged in the first three positions in $3!$ ways. Number of such numbers = $3! = 6$. Case 4: The last two digits are 52. The remaining 3 digits (1, 3, 4) can be arranged in the first three positions in $3!$ ways. Number of such numbers = $3! = 6$. The total number of 5-digit numbers divisible by 4 (and thus also even) is $6 + 6 + 6 + 6 = 24$. Step 3: Calculate the conditional probability. The probability that a randomly chosen 5-digit number is divisible by 4, given that it is even, is the ratio of the number of 5-digit numbers that are divisible by 4 (which are also even, as established in Step 2) to the total number of even 5-digit numbers. $$ P(\text{divisible by 4} | \text{even}) = \frac{\text{Number of 5-digit numbers divisible by 4}}{\text{Number of even 5-digit numbers}} $$ $$ P(\text{divisible by 4} | \text{even}) = \frac{24}{48} = \frac{1}{2} $$