Place blue, white and red balls in a urn. Prove that the probability that the ith ball taken from the box without replacement is:
$$ P(X_i=r) = \frac{r}{r+b+w}$$
There is a way to prove this result mathematically? Its true for $P(X_2 =r)$ (using total probability law), but how can I prove and interpret this result?
Thanks!
Best Answer
You may approach this in more than one way.
Let $n = r+b+w$.
"Direct way":
Each of the $n$ balls are equally likely to appear at the $i$-th position. There are $r$ red balls. So, you get $$P(X_i = r) = \frac{r}{n}= \frac{r}{r+b+w}$$
"Counting way:"
A possible way of counting is as follows:
All together: $$P(X_i = r)=\frac{\color{red}{r} \cdot (n\color{red}{-1})!}{n!}= \frac{\color{red}{r}}{n} = \frac{r}{r+b+w}$$