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:


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


1 Answer 1


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*}


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.