Answer to Question #195980 in Statistics and Probability for Zub

Question

The latest M1 Macbook Air and Macbook Pro are available for sale at the Apple Store in Gulshan. 70% of the people who want to buy a new Macbook prefer the Macbook Air, while the rest prefer the Macbook Pro. Suppose 16 customers were randomly selected among those shopping for Macbooks on a specific day.

a) What is the mean value and the standard deviation of the number of people who want to purchase a Macbook Air? [2 marks]

b) What is the probability that the number of people who want a Macbook Air is more than two standard deviations away from the mean value? [4 marks]

c) The Apple Store only has 11 Macbook Airs and 14 Macbook Pros in stock. If the 16 people come in one after another to purchase a Macbook, what is the probability that all 16 will get the type of Macbook they want? [4 marks]

Answer 1

a.) We have,

$n = 16$

$p= 0.7$

$q = 0.3$

Mean value $= np = 16 \times 0.7 = 11.2$

Standard Deviation $= \sqrt{npq} = \sqrt{16 \times 0.7 \times 0.3} = 1.83$

b.) The probability that the number of people who want a Macbook Air is more than two standard deviations away from the mean value

$= P(X<\mu-2\sigma)+ P(X>\mu+2\sigma)$

$= P(X<11.2-2(1.83))+P(X>11.2+2(1.83))$

$= P(X<7.54)+P(X>14.86)$

$= P(Z<\dfrac{7.54-11.2}{1.83})+P(Z>\dfrac{14.86-11.2}{1.83})$

$= P(Z<-2)+P(Z>2)$

$= 0.0228+0.0228 = 0.0456$

c.)  The probability that all 16 will get the type of Macbook they want $= \dfrac{^{16}C_{16}}{^{27}C_{16}}$