Final answer:
The probability that a randomly chosen four-digit number formed from the digits 1, 2, 3, and 5 without repetition is divisible by 4 is 0.1667 when rounded to four decimal places.
Step-by-step explanation:
To find the probability that a four-digit number formed from the digits 1, 2, 3, and 5 without repetition is divisible by 4, we need to know the divisibility rule for 4: a number is divisible by 4 if its last two digits form a number that is divisible by 4.
Since we are forming four-digit numbers, we need only consider combinations of the last two digits. There are six possible two-digit combinations from these digits: 12, 32, 52, 24, 14, and 34. Out of these, only 12 and 32 are divisible by 4.
These two combinations can be at the end of a four-digit number, and we can arrange the remaining two digits in the front in two different ways each (since they are not repeated).
Therefore, we have 2 (choices for the last two digits) × 2 (ways to arrange the first two digits) = 4 possible four-digit numbers that are divisible by 4. In total, there are 4! or 24 possible four-digit numbers that can be formed using the digits 1, 2, 3, and 5 without repetition.
Hence, the probability is calculated as follows:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 4 / 24
Probability = 0.1667 (rounded to four decimal places)