Probability that the bus will arrive within 4-5 minutes

Question

The bus runs at intervals of 10 minutes, and at a random moment you come to a stop. What is the probability that the bus will arrive within 4-5 minutes? My textbook says the answer is $\frac{1}{10}$ but i don't understand why. I understand why probability that the bus will arrive within 1 minute is $\frac{1}{10}$, cause there are 10 outcomes of which one is positive. But in case of 4-5 minutes there are 2 positive cases, so it must be $\frac{2}{10}=\frac{1}{5}$. Where I'm making mistake?

Answer 1

You must focus on the single interval $[4,5]$ and not on the two points $\{4,5\}$.

There are $10$ almost disjoint and covering subintervals of the form $[i,i+1]$ (where $i$ is an integer) for $[0,10]$ and they are equiprobable when it concerns the question whether the arrival time of the bus will fall in them.

Answer 2

For what it's worth, I'm consistently getting a mean of about $\frac1{10}$ with a simulation:

# Number of busses
N <- 10000
# Bus arrivals
B <- rep(10, N)
# Your arrivals
X <- B - runif(N, 0, 10)
# Output results
hist(X)
cat(length(which(X>4 & X< 5)) / N)