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, Onkyo, Pioneer, Sony, Sherwood

Compact disc player: Onkyo, Pioneer, Sony, Technics

Speakers: Boston

Cassette deck: Onkyo, Sony, Teac, Technics
(b) In how many ways can components be selected if both the receiver and the compact disc player are to be Sony?


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


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


(e) 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?


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

Answer 1

Receiver: Kenwood, Onkyo, Pioneer, Sony, Sherwood

Compact disc player: Onkyo, Pioneer, Sony, Technics

Speakers: Boston

Cassette deck: Onkyo, Sony, Teac, Technics

(b)

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

i.e

Receiver : Sony

Compact disc player : Sony

Speakers : Boston

Cassette deck : Onkyo, Sony, Teac, Technics

Number of ways a receiver can be selected = 1

Number of ways a Compact disc player can be selected = 1

Number of ways a speaker can be selected = 1

Number of ways a Cassette deck can be selected = 4

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

= 1 x 1 x 1 x 4 =4

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

(c) In how many ways can components be selected if none is to be Sony

i.e

Receiver: Kenwood, Onkyo, Pioneer, Sherwood

Compact disc player: Onkyo, Pioneer, Technics

Speakers: Boston

Cassette deck: Onkyo, Teac, Technics

Number of ways a receiver can be selected = 4

Number of ways a Compact disc player can be selected = 3

Number of ways a speaker can be selected = 1

Number of ways a Cassette deck can be selected = 3

Number of ways can components be selected if none is to be Sony = 4 x 3 x 1 x 3 = 36

Number of ways can components be selected if none is to be Sony = 36

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

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

= Total Number of ways a selection be made(without any restrictions) - Number of ways can components be selected if none is to be Sony

Selection be made without any restrictions

Receiver: Kenwood, Onkyo, Pioneer, Sony, Sherwood

Compact disc player: Onkyo, Pioneer, Sony, Technics

Speakers: Boston

Cassette deck: Onkyo, Sony, Teac, Technics

Number of ways a receiver can be selected = 5

Number of ways a Compact disc player can be selected = 4

Number of ways a speaker can be selected = 1

Number of ways a Cassette deck can be selected = 4

Total Number of ways a selection be made(without any restrictions) = 5 x 4 x 1 x 4 = 80

From (c) :Number of ways can components be selected if none is to be Sony = 36

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

= Total Number of ways a selection be made(without any restrictions) - Number of ways can components be selected if none is to be Sony

= 80 - 36 = 44

Number of ways a selection be made if at least one Sony component is to be included = 44

(e) If someone flips switches on the selection in a completely random fashion,

probability that the system selected contains at least one Sony component

= Number of ways the system selected contains at least one Sony component / Total Number of ways a selection be made(without any restrictions)

From (d) ,

Number of ways a selection be made if at least one Sony component is to be included = 44

Total Number of ways a selection be made(without any restrictions) = 80

probability that the system selected contains at least one Sony component

= Number of ways the system selected contains at least one Sony component / Total Number of ways a selection be made(without any restrictions)

= 44/80 = 11/20 = 0.55

If someone flips switches on the selection in a completely random fashion, probability that the system selected contains at least one Sony component = 0.55

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

Probability that the system contains exactly one Sony component

The ways that the system contains exactly one Sony component :

Option 1 : (Receiver Sony)

Receiver: Sony

Compact disc player: Onkyo, Pioneer , Technics

Speakers: Boston

Cassette deck: Onkyo, Teac, Technics

Number of ways the system selected as per option 1 = 1x3x1x3 =9

Option 2: (Compact disc player Sony)

Receiver: Kenwood, Onkyo, Pioneer, Sherwood

Compact disc player: Sony

Speakers: Boston

Cassette deck: Onkyo, Teac, Technics

Number of ways the system selected as per option 2 = 4 x 1 x 1 x 3 =12

Option 3 : (Cassette deck:Sony)

Receiver: Kenwood, Onkyo, Pioneer, Sherwood

Compact disc player: Onkyo, Pioneer, Technics

Speakers: Boston

Cassette deck: Sony

Number of ways the system selected as per option 3 = 4 x 3 x 1 x 1 =12

Number of ways that the system contains exactly one Sony component

= Number of ways the system selected as per option 1 + Number of ways the system selected as per option 2 + Number of ways the system selected as per option 3

=9+12+12 = 33

Probability that the system contains exactly one Sony component

= Number of ways that the system contains exactly one Sony component / Total Number of ways a selection be made(without any restrictions)

= 33/80 = 0.4125

Probability that the system contains exactly one Sony component = 0.4125