Question

Four balls are drawn from a bag containing 5 black, 6 white and 7 red balls. Let X is no. of white balls drawn. find the probability mass function or probability distribution of X.

Answer 1

there are 18$C_{4}$ ways to selecting 4 balls from 5 black, 6 white, 7 red balls
we can select x from 6 white balls and 4-x from remaining balls in (6$C_{x}$)(12$C_{4-x}$) ways. hence
f(x) = (6$C_{x}$)(12$C_{4-x}$) / 18$C_{4}$   x = 0,1,2,3,4
I hope you can understood. if not please ask.

Answer 2

There are four while balls (W) and 12 other balls (O) in the bag. Four balls are drawn one after another, let us say. The drawing of 4 balls are independent  events and mutually exclusive. So probabilities can be multiplied.

X = event that X number of the four drawn balls are white balls.

P(X=0) = all drawn balls are non-white.

$P(X=0) = \frac{12}{18} * \frac{11}{17} * \frac{10}{16} * \frac{9}{15} = 0.1617 \\ \\ P(X=1) = 4 * \frac {6* 12*11*10} {18*17*16*15} = 0.4314 \\ \\ P(X=2) = 6 * \frac{6*5* 12*11}{18*17*16*15} = 0.3235 \\ \\ P(X=3) = 4 * \frac{6*5*4*12}{18*17*16*15} = 0.0784 \\ \\ P(X=4) = \frac{6*5*4*3}{18*17*16*15} = 0.0049$