Question

In this problem, we will analyze basic probability concepts. For each calculation provide

your answer and explain how you made the calculation. You may show a formula, an Excel formula, a
table, or describe textually. Your goal is to explain clearly and succinctly.

Do the following:

• Recently Apple announced the new iPhone XS, and the iPhoneXS Max. They are offered with a
number of options including

o Three Colors: Silver, Space Grey and Gold

o Three Capacities: 64 GB, 256GB and 512 GB

o 4 Carriers (AT&T, Spring, Verizon, T-Mobile) and carrier free

• At the same time, they announced the iPhone XR with the following options:

o Six Colors: White, Black, Blue, Yellow, Coral, Red

o Three Capacities: 64 GB, 256GB and 512 GB

o 4 Carriers and carrier free

• Based on this information calculate the total number of new configurations
available.

• Calculate the total number of configurations available ignoring the carrier.

• A retailer wants to display 5 phones in an interactive display.

o Calculate the total number of ways the retailer can select phone configurations for the
display, ignoring carrier and order.

o Recalculate the number of ways the phones can be displayed if order of display matters.

• Assume that a battery defect effects 1% of the phones. Calculate the following probabilities
for the 5 display phones:

o None of the phones have the defect

o Exactly one of the phones has the defect.

o The set of 5 phones has 1 or less defects.

• Now assume the retailer wants to stock 10 phones of each configuration for the no carrier
option and 25 for the other 4 carriers. Based on this population, calculate the
following:

o The total number of phones to stock

o The probability a randomly selected iPhone is Silver.

o The probability a randomly selected iPhone is an iPhone XS Max given

that it is Gold.

Answer 1

iPhone XS

3 Colors: Silver, Space Grey and Gold

3 Capacities: 64 GB, 256 GB and 512 GB

4 Carriers (AT&T, Spring, Verizon, T-Mobile) and carrier free

iPhone XS Max

3 Colors: Silver, Space Grey and Gold

3 Capacities: 64 GB, 256 GB and 512 GB

4 Carriers (AT&T, Spring, Verizon, T-Mobile) and carrier free

iPhone XR

6 Colors: White, Black, Blue, Yellow, Coral, Red

4 Capacities: 64 GB, 256 GB and 512 GB

4 Carriers and carrier free

.

Total number of new configurations available = 3$\times$3$\times$5 + 3$\times$3$\times$5 + 6$\times$4$\times$5 = 210

Total number of configurations available ignoring the carrier = 3$\times$3 + 3$\times$3 + 6$\times$4 = 42

Total number of ways the retailer can select 5 phone configurations for the display, ignoring carrier and order  

42 850668

Total number of ways 5 phone configurations can be displayed ignoring carrier if order of display matters

= (42)₅ = 102080160

Probability of a battery defect = 1% = 0.01

Probability that none of the phones have the defect

(. C) (0.01)90.99" = 0.950990049 (0.01)(0.90.950990049

Probability that exactly one of the phones has the defect

(0.01)(0.99) 0.0480298

Probability that the set of 5 phones has 1 or less defects.

= Probability that none of the phones have the defect + Probability that exactly one of the phones has the defect

= 0.950990049 + 0.0480298 = 0.99901985

The retailer wants to stock 10 phones of each configuration for the no carrier option and 25 for the other 4 carriers.

Total number of phones to stock = 10$\times$(3$\times$3 + 3$\times$3 + 6$\times$3) + 25$\times$4$\times$(3$\times$3 + 3$\times$3 + 6$\times$3) = 3960

Probability a randomly selected iPhone is Silver

= (No. of phones of color Silver)/(Total number of phones to stock) = (10$\times$3 + 10$\times$3 + 25$\times$4$\times$3 + 25$\times$4$\times$3)/3960 = 1/6

Probability a randomly selected iPhone is an iPhone XS Max given that it is Gold

= (Probability a randomly selected iPhone is a Gold iPhone XS Max)/(Probability a randomly selected iPhone is Gold)

(Number of Gold iPhone XS Max)/(Total number of phones) (Number of Gold iPhone) /(Total number of phones)

(10 × 3 25 × 4 × 3)/3960 10 × 3 25 × 4 × 3 25 × 4 ×3)/3960 (10 × 3