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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?

User Mitsuruog
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1 Answer

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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}

User Leniency
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