MHB Finding a formula for a problem

[Fermat1]

There are n birds which need to be put back in a cage.

In a specific 10 minute period each bird will be put back at some random point in time.

Find a formula for the probability that m animals from n are put back at the same time, where the 'same time' means in the same 1 second interval. The intervals are (0,1),(1,2) etc

My idea. Let's take any particular interval. Then I think it boils down to a binomial situation with n trials and probability of success 1/600.

So for any particular interval, the probability of m from n being in the interval is

nCm*(1/600)^m*(599/600)^(n-m)

because there are 600 intervals we should multiply the above by 600

Is this correct?

Thanks

 

[Fermat1]

please help. I've shown my work so I think its deserving.

 

[Jameson]

Hi Fermat,

Assuming that each interval is equally likely, then yes I agree with you.

As you said the probability of $m$ animals being put back in the interval (0,1) is $${n \choose m}\left( \frac{1}{600} \right)^m \left(\frac{599}{600} \right)^{n-m}$$. It's same probability for (1,2),(2,3)...(599,600). We have 600 intervals so multiply the above probability by 600.