# Expectation of balls in a jar

Trying to solve this question:

A jar contains 10 balls numbered 1,2,3,...,10. We draw 15 balls from the jar, one after the other, with replacement. Let N denote the number of distinct numbers drawn.

For example, if all numbers are drawn except 1 and 2, then N = 8.

Find the expected value of N.

I've tried to solve it by calculating the summation of this:

(x)(10Cx)(x/10)^15

Where x goes from 1 till 10. I got 45.855 but it's wrong apparently.

Let $$E_i$$ be the event that the number $$i$$ has been seen at least once in the drawn sequence of the numbers of balls where $$i=1,\dots,10$$, and $$1_{E_i}$$ be the indicator random variable of the event $$E_i$$.
Now, Let $$N$$ denote the number of distinct numbers drawn.
We have $$N= \sum_ {i=1}^{10} 1_{E_i}$$ and thus the expected value of $$N$$, \begin{align*} \mathbb{E} \left( N \right) & = \sum_ {i=1}^{10} \mathbb{E} \left( 1_{E_i} \right)\\ & = \sum_ {i=1}^{10} \mathbb{P} \left( {E_i} \right)\\ & = \sum_ {i=1}^{10} \left(1- \mathbb{P} \left( {E_i}^c \right)\right)\\ & = \sum_ {i=1}^{10} \left(1- \left( \frac{9}{10} \right)^{15} \right) \\ & = 10 \times \left(1- \left( \frac{9}{10} \right)^{15} \right) \\ & \approx 7.941 \end{align*}