Question

A stereo store is offering a special price on a complete set of components (receiver, compact disc player, speakers, cassette deck). A purchaser is offered a choice of manufacturer for each component:

Receiver: Kenwood, Sony, Sherwood

Compact disc player: Onkyo, Pioneer, Sony, Technics

Speakers: Boston

Cassette deck: Onkyo, Sony, Teac, Technics

A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions.

In how many ways can one component of each type be selected?

In how many ways can components be selected if both the receiver and the compact disc player are to be Sony?

In how many ways can components be selected if none is to be Sony?

In how many ways can a selection be made if at least one Sony component is to be included?

If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component?

What is the probability that the system contains exactly one Sony component?

Answer 1

Receiver: Kenwood, Sony, Sherwood: Number of choices for receivers = 3

Compact disc player: Onkyo, Pioneer, Sony, Technics : number of choices for Compact disc player = 4

Speakers: Boston : number of choices for speakers = 1

Cassette deck: Onkyo, Sony, Teac, Technics : number of choices for cassette deck = 4

In how many ways can one component of each type be selected?

Number of ways one component of each type be selected

Number of ways of selecting one receivers from 3 receivers x

Number of ways of selecting one Compact disc player from 4 Compact disc players x

Number of ways of selecting one speaker from 1 speaker x

Number of ways of selecting one Cassette deck from 4 cassette deck

= 3 x 4 x 1 x 4 = 48

Number of ways one component of each type be selected =48

In how many ways can components be selected if both the receiver and the compact disc player are to be Sony?

Sony receiver (1)

Sony compact disc player

Number of ways of selecting one speaker from 1 speaker x

Number of ways of selecting one Cassette deck from 4 cassette deck

= 1 x 1 x 1 x 4 = 4

Number of ways components be selected if both the receiver and the compact disc player are to be Sony = 4

In how many ways can components be selected if none is to be Sony?

Number of ways of selecting one receivers from 2 non-Sony receivers x

Number of ways of selecting one Compact disc player from 3-non-Sony Compact disc players x

Number of ways of selecting one speaker from 1 speaker x

Number of ways of selecting one Cassette deck from 3 non-Sony cassette deck

= 2 x 3 x 1 x 3 = 18

Number of ways components be selected if none is to be Sony=18

In how many ways can a selection be made if at least one Sony component is to be included?

Number of ways a selection can be made if at least one Sony component is to be included

= Total Number of ways one component of each type be selected(without any restrictions) - Number of ways components be selected if none is to be Sony

= 48-18 =30

Number of way a selection can be made if at least one Sony component is to be included =30

If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component?

Probability that the system selected contains at least one Sony component

= Number of way a selection can be made if at least one Sony component is to be included /Total Number of ways one component of each type be selected

=30/48 = 5/8=0.625

Probability that the system selected contains at least one Sony component = 0.625

Probability that the system contains exactly one Sony component

Number of way a selection can be made that contains exactly one Sony component

=

[Sony receiver x

Number of ways of selecting one Compact disc player from 4 Compact disc players x

Number of ways of selecting one speaker from 1 speaker x

Number of ways of selecting one Cassette deck from 4 cassette deck] (1x4x1x4=16)

+

[Number of ways of selecting one receivers from 3 receivers x

Sony Compact disc player

Number of ways of selecting one speaker from 1 speaker x

Number of ways of selecting one Cassette deck from 4 cassette deck] (3x1x1x4=12)

+

[Number of ways of selecting one receivers from 3 receivers x

Number of ways of selecting one Compact disc player from 4 Compact disc players

Number of ways of selecting one speaker from 1 speaker x

Sony Cassette deck ](3x4x1x1=12)

= (1x4x1x4)+(3x1x1x4)+(3x4x1x1)=16+12+12=40

Number of way a selection can be made that contains exactly one Sony component = 40

Probability that the system contains exactly one Sony component

= Number of ways a selection can be made that contains exactly one Sony component /Total Number of ways one component of each type be selected

= 40/48 = 5/6=0.8333

Probability that the system contains exactly one Sony component =5/6 =0.8333