# Find the expectation

A box contain $A$ white and $B$ black balls and $C$ balls are drawn, then the expected value of the number of white balls drawn is ?

The answer is $\large \frac{ca}{a+b}$. How to approach this one?

-
 FYI: The number of white balls is said to have a hypergeometric distribution: en.wikipedia.org/wiki/Hypergeometric_distribution. – Mike Spivey Jan 27 '11 at 17:09

These calculations are often made clearer by using indicator random variables. For $1\leq i\leq C$, define $Z_i$ by setting $Z_i=1$ if the $i$th ball drawn is white and $Z_i=0$ otherwise. Then the total number of white balls is $\sum_{i=1}^C Z_i$ and by linearity of expectation we have $$E(\mbox{ white balls })=E\left(\sum_{i=1}^C Z_i\right)=\sum_{i=1}^C E(Z_i).$$

The expectation of an indicator random variable is just the probability of the event it indicates, so $E(Z_i)=P(i\mbox{th ball is white})=A/(A+B)$.

Therefore we find that $$E(\mbox{ white balls })=\sum_{i=1}^C E(Z_i)={C A\over A+B}.$$

Notice that we did not need to calculate the probability of getting one, two, three, etc. white balls.

-

For those interested, just to elucidate on the "non-smart" method of attacking this problem, the probability of drawing exactly $k$ white balls in $C$ draws is $$\frac{ {A \choose k}{B \choose C-k}} { {A+B \choose C} }$$ and so the expected number of white balls is given by

$$\sum_{k=0}^C \frac{ k {A \choose k}{B \choose C-k}} { {A+B \choose C} } = \frac{AC}{A+B}.$$

-

The formula for the expected value of number of white balls is

$E($number of white drawn$) = 1P($1 white ball in C balls drawn $) + 2P($2 white balls in C balls drawn $) + \dots + C P($ C white balls drawn $)$.

Now you need to find the probabilities. For example

$P($ C white balls drawn$) = P($first ball is white$) * P($ second ball is white given first ball was white$) * P($ third ball is white given balls 1 and 2 were white$) * \dots * P($last ball white given all C-1 balls were white$)$

-
 This is just an objective type question ... are you sure that we don't have any other smart method ? – Quixotic Jan 27 '11 at 14:10 @Debanjan: No, there might be cleverer ways of computing this expectation value. – Matt N. Jan 27 '11 at 14:28