Find probability of an event.

Question

Let there be $100$ balls in an urn out of which $50$ are red and $50$ are green. Let $A$ be the event of drawing $75$ balls from the urn in which $n$ balls are red in colour, where $25 \leq n \leq 50.$ Find $\Bbb P(A).$

Note $:$ All the red (resp. green) balls are indistinguishable.

I know the result if the balls are distinguishable in which case $$\Bbb P(A) = \frac {\binom {50} {n} × \binom {50} {75-n}} {\binom {100} {75}}.$$

How can I solve this question for indistinguishable balls? Any help will be appreciated.

Thanks in advance.

Answer 1

There is no distinction between distinguishable and indistinguishable balls in probability issues. Saying "indistinguishabe" you just mean that any distinction between the balls is ignored. It becomes obvious if you number the balls. Obviously the probability to draw $n$ red balls does not depend on the fact whether the balls are numbered or not.

It can be different for really indistinguishable objects such as quantum particles. This is the reason why they do not obey the classical Boltzmann statistics. But for classical objects such as balls this plays no role.