Question

A denim company sells its jeans both online and at the retail store. Assume that 80% of the company's sales are retail, and 20% of sales are online. A. What's the probability that all of the next four pairs of jeans are sold online? B. What's the probability that 3 out of the next 4 parts of jeans are sold online? C. Use your answers from parts (a and b) to derive a formula for p(x), the probability distribution of the binomial random variable x, the number of the next 4 pats of jeans sold online **PLEASE SHOW CLEAR CALCULATIONS SO I CAN UNDERSTAND HOW TO WORK OTHER PROBLEMS**

Answer 1

Given That

company's sales are retail = 80%

company's sales are online = 20%

P(pair is sold online) = 20% = 0.2

A. The probability that all of the next four pairs of jeans are sold online

P(all of the next 4 pairs of jeans are sold online) = (0.2)⁴

= 0.2 x 0.2 x 0.2 x 0.2

= 0.0016

P (all of the next 4 pairs of jeans are sold online) = 0.0016

B. The probability that 3 out of the next 4 parts of jeans are sold online

P (3 out of the next 4 parts of jeans are sold online) = ⁴C₃ x 0.2 x 0.2 x 0.2 x 0.8

  ⁴C₃ means combinations of 3 out of four parts = 1

3 are sold online = 0.2*0.2*0.2

1 are sold retail = 0.8

P (3 out of the next 4 parts of jeans are sold online) = ⁴C₃ x 0.2³ x 0.8¹

= 1* 0.2*0.2*0.2*0.8

= 0.0064

P (3 out of the next 4 parts of jeans are sold online) = 0.0064

C. Derive formula for p(x):

the probability distribution of the binomial random variable x

the number of the next 4 pats of jeans sold online

General formula for probability distribution of the binomial

P(x) = ⁿC_x * P^x * q^{(n-x) ............. (1)}

where

n is No of parts

x is random variable

P is probability of online sale

q is probability of retail sale or 1-p

^{here n = 4}

x is x

p = 0.2

q = 0.8

substitute values in eq (1)

P(x) = 4C_x * 0.2^x * 0.8^(4-x)