# A sample of 4 bulbs is drawn at random with replacement from a lot of 30 bulbs which includes 6 defective bulbs. Find the probability distribution of the number of defective bulbs. - Mathematics and Statistics

Sum

A sample of 4 bulbs is drawn at random with replacement from a lot of 30 bulbs which includes 6 defective bulbs. Find the probability distribution of the number of defective bulbs.

#### Solution

Let X denote the number of defective bulbs.

∴ Possible values of X are 0, 1, 2, 3, 4.

Let P(getting a defective bulb) = p = (6)/(30) = (1)/(5)

∴ q = 1 – p = 1 - (1)/(5) = (4)/(5)

∴ P(X = 0) = P(no defective bulb)

= qqqq = q4 = (4/5)^4

P(X = 1) = P(one defective bulb)

= qqqp + qqpq + qpqq + pqqq

= 4pq3

= 4 xx (1)/(5) xx (4/5)^3 = (4/5)^4

P(X = 2) = P(two defective bulbs)

= ppqq + pqqp + qqpp + pqpq + qpqp + qppq

= 6p2q2

= 6(4/5)^2(1/5)^2

P(X = 3) = P(three defective bulbs)

= pppq + ppqp + pqpp + qppp

= 4qp3

= 4(4/5)(1/5)^3

P(X = 4) = P(four defective bulbs)

= pppp = p4 = (1/5)^4

Probability distribution of X is as follows:

 X 0 1 2 3 4 P(X = x) (4/5)^4 (4/5)^4 6(4/5)^2(1/5)^2 4(4/5)(1/5)^3 (1/5)^4
Concept: Random Variables and Its Probability Distributions
Is there an error in this question or solution?

#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Probability Distributions
Exercise 8.1 | Q 1.06 | Page 141