probability

Question

A total of n balls are sequentially and randomly chosen, without replacement, from an urn containing r red and b blue balls (n … r + b). Given that k of the n balls are blue, what is the conditional probability that the first ball chosen is blue?

Answer 1

an urn containing r red and b blue balls

So, it can contain ^rC₁ ball or ^bC₁ balls

Now,  among n balls k balls are blue, (n-k) balls are red

So, 1st urn can contain $\frac{kC1*bC1}{kC1*bC1+(n-k)C1*rC1}$

Answer 2

To choose $n$ ball from $(r+b)$ balls can be done in $\binom{r+b}{n}$

now, k ball out of those n chosen balls are guaranteed to be blue, so to make sure that first ball is blue we can choose our first ball from these k balls in $\binom{k}{1}$ and remaining $(n-1)$ ball in $\binom{r+b-k}{n-1}$ ways.

$Probability=$$\frac{\binom{k}{1}\binom{r+b-k}{n-1}}{\binom{r+b}{n}}$