Question

There are 10 balls in an urn, numbered from 1 to 10. If 5 balls are selected at random and their numbers are added, what is the expected value of the total?

Answer 1

Let
`B_i` denote the value on the
`i`-th drawn ball. We want to find the expectation of
`S=B_1+B_2+B_3+B_4+B_5`, which by linearity of expectation is

`E[S]=E\left[\displaystyle\sum_(i=1)^5B_i\right]=\sum_(i=1)^5E[B_i]`

(which is true regardless of whether the
`X_i` are independent!)

At any point, the value on any drawn ball is uniformly distributed between the integers from 1 to 10, so that each value has a 1/10 probability of getting drawn, i.e.

`P(X_i=x)=\begin{cases}\frac1{10}&\text{for }x\in\{1,2,\ldots,10\}\\0&\text{otherwise}\end{cases}`

and so

`E[X_i]=\displaystyle\sum_(i=1)^(10)x\,P(X_i=x)=\frac1{10}\frac{10(10+1)}2=5.5`

Then the expected value of the total is

`E[S]=5(5.5)=\boxed{27.5}`