Question

# Box 1 contains 4 red balls, 5 green balls and 1 yellow ball. Box 2 contains...

Box 1 contains 4 red balls, 5 green balls and 1 yellow ball.

Box 2 contains 3 red balls, 5 green balls and 2 yellow balls.

Box 3 contains 2 red balls, 5 green balls and 3 yellow balls.

Box 4 contains 1 red ball, 5 green balls and 4 yellow balls.

Which of the following variables have a binomial distribution?
(I) Randomly select three balls from Box 1 with replacement. X = number of red balls selected

(II) Randomly select one ball from each of the four boxes. X = number of yellow balls selected

(III) Randomly select three balls from Box 1 without replacement. X = number of red balls selected

(IV) Randomly select one ball from each of the four boxes. X = number of green balls selected

(I)

Since ball are drawn with replacement so probability of getting a red ball is p = 4/10 each time.

It is binomial distrbution with parameters n=3 and p= 0.40.

(II)

Since probability of getting yellow ball from each box is different so it is not binomial.

(iii)

Since ball are drawn with replacement so it is not binomial. In this case probability of getting a red ball is not same each time.

(iv)

The probability of getting green ball from each is 5/10 = 0.20

So here X has binomial distribution with parameter n = 1+1+1+1= 4 and p=0.20.