Question

The manager of Toy World knows that the probability an electronic game will be returned to the store is 0.22. If 54 games are sold in a given week, determine the probabilities of the following events. (Round your answers to four decimal places.)

(a) No more than 12 games will be returned.


(b) At least 8 games will be returned.


(c) More than 5 games but fewer than 14 games will be returned.

Answer 1

n = 54

p = 0.22

np = 54 * 0.22 = 11.88

n(1 - p) = 54 * 0.78 = 42.12

As np > 5 and n(1 - p) > 5, so we can use normal approximation to the binomial distribution.

= np = 54 * 0.22 = 11.88

= sqrt(54 * 0.22 * 0.78) = 3.044

a) P(X < 12)

= P(X < 12.5)

= P((X - )/< (12.5 - )/)

= P(Z < (12.5 - 11.88)/3.044)

= P(Z < 0.20)

= 0.5793

b) P(X > 8)

= P(X > 7.5)

= P((X - )/< (7.5 - )/)

= P(Z < (7.5 - 11.88)/3.044)

= P(Z < -1.44)

= 0.0749

c) P(5 < X < 14)

= P(6 < X < 13)

= P(5.5 < X < 13.5)

= P((5.5 - )/< (X - )/< (13.5 - )/)

= P((5.5 - 11.88)/3.044 < Z < (13.5 - 11.88)/3.044)

= P(-2.10 < Z < 0.53)

= P(Z < 0.53) - P(Z < -2.10)

= 0.7019 - 0.0179

= 0.684