an urn contains 100 balls: 20 white, 30 black and 50 gray. SUpposed that 20 balls are chosen at random and without replacement. Let W,B, and G be the number of balls, respectibely. Calculate the probability mass function of B,W, and G
an urn contains 100 balls: 20 white, 30 black and 50 gray. SUpposed that 20 balls are chosen at random and without replacement. Let W,B, and G be the number of balls, respectibely. Calculate the probability mass function of B,W, and G
You have a multivariate hypergeometric distribution:
Pr(W = w, B = b, G = g) $\displaystyle = \frac{ {20 \choose w} {30 \choose b} {50 \choose g}}{{100 \choose 20}}$
where w + b + g = 20.
You could also build in the w + b + g = 20 restriction and write the pmf as:
Pr(W = w, B = b, G = g) $\displaystyle = \frac{ {20 \choose w} {30 \choose b} {50 \choose {20 - w - b} }}{{100 \choose 20}}$.
And, I hate to say it, but the jury question you posted a while ago will also follow a multivariate hypergeometric distribution since (I assume) there's no replacement after each juror is selected. The card one too - if there's no replacement. I've added a reply noting this to each of those threads.