You have a lottery number with ten slots. On each slot is an equally probably number from 0 to 1. You are paid the maximum number of the ten slots. What is the expected payout of the lottery number.
My strategy was to use the integral definition for expected value. The probability $p(x)$ that a single value $x$ is the highest of all the ten slots means that all the other values must fall within the range from $0$ to $x$, which happens with probability of $x^9$. Since there are ten different spots that this happens is $10x^9$. Then using the definition of expected value, we get $\int_0^1 x10x^9 \,dx$ = $\frac{10}{11}$.
Does this look correct?
Based on your description of the problem, it is expected that the expectation value will be 5. Since it is equal probability to choose a number between 0 and 1 for each box, the expectation value in each box is 1/2. Thus summing over 10 boxes would make the value 5. More rigorously, you can represent the expectation value for each box as an integral then sum over the boxes $$E(X) = \sum_{n=1}^{10}\int_0^1x\,dx=\sum_{n=1}^{10}\frac{1}{2}=5$$