Question

Four balls are to be drawn without replacement from a box containing 8 red and 4 white balls. If X denotes the number of red ball drawn, find the probability distribution of X.

Answer 1

Since 4 balls have to be drawn, therefore, X can take the values 0, 1, 2, 3, 4.

P(X = 0) = P(no red ball) = P(4 white balls)

= `(""^4"C"_4)/(""^12"C"_4) = 1/495`

P(X = 1) = P(1 red ball and 3 white balls)

= `(""^8"C"_1 xx ""^4"C"_3)/(""^12"C"_4) = 32/495`

P(X = 2) = P(2 red balls and 2 white balls)

= `(""^8"C"_2 xx ""^4"C"_2)/(""^12"C"_4) = 168/495`

P(X = 3) = P(3 red balls and 1 white ball)

= `(""^8"C"_3 xx ""^4"C"_1)/(""^12"C"_4) = 224/495`

P(X = 4) = P(4 red balls) 

= `(""^8"C"_4)/(""^12"C"_4) = 70/495` 

Thus the following is the required probability distribution of X

X	0	1	2	3	4
P(X)	`1/495`	`32/495`	`168/195`	`224/495`	`70/495`