A pond contains red and golden fish. There are $3000$ red and $7000$ golden fish, of which $200$ and $500$ respectively, are tagged. Find the probability that a random sample of $100 $ red and $200$ golden fish will show $15$ and $20$ tagged fish, respectively
My input :
$A$: Choosing $15$ tagged redfish
$B$: Choosing $20$ tagged golden fish
$P$($A$ and $B$)=$P(A)P(B)$ $\ \ \ \ \ \because $ Given its random sample.
$\dfrac{\binom{100}{15}\binom{2800}{85}}{{3000}\choose{100}}\cdot\dfrac{\binom{500}{20}\binom{6500}{180}}{{7000}\choose{200}}$
I think I have done something wrong. I don't have the answer to this question in my textbook so please someone check it.
$$ \frac{\binom{3000 - 200 }{100 - 15}\binom{200 }{15}}{\binom{3000}{100}} \times\frac{\binom{7000 - 500 }{200 - 20}\binom{500 }{20}}{\binom{7000}{200}} $$