One digit of the decimal representation of $\frac{5}{7}$ is randomly selected. What is the probability that the digit is a 4? Express your answer as a common fraction.
I need to find the decimal representation of $\frac{5}{7}$ first, and then see how often 4 appears in it. I can use long division to find the decimal representation. I divide 5 by 7 and get a quotient of 0 and a remainder of 5. I bring down a 0 and divide 50 by 7 and get a quotient of 7 and a remainder of 1. I bring down another 0 and divide 10 by 7 and get a quotient of 1 and a remainder of 3. I bring down another 0 and divide 30 by 7 and get a quotient of 4 and a remainder of 2. I bring down another 0 and divide 20 by 7 and get a quotient of 2 and a remainder of 6. I bring down another 0 and divide 60 by 7 and get a quotient of 8 and a remainder of 4. I bring down another 0 and divide 40 by 7 and get a quotient of 5 and a remainder of 5. I notice that I have reached the same remainder as before, so the decimal representation of $\frac{5}{7}$ is 0.714285 and it repeats indefinitely. The digit 4 appears once in every cycle of 6 digits, so the probability of randomly selecting a 4 is $\frac{1}{6}$.

