Question

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

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:

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

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? Exactly one Sony component?

Short answer

1. \({\rm{240}}\) ways
2. \({\rm{12}}\) ways
3. \({\rm{108}}\) ways
4. \({\rm{132}}\) ways
5. At least one sony component

\({\rm{P( }}\)at least one sony component\({\rm{) = }}\frac{{{\rm{11}}}}{{{\rm{20}}}}{\rm{ = 0}}{\rm{.55 = 55\% }}\)

Exactly one sony cokmponent

\({\rm{P(}}\)exactly one sony component\({\rm{) = }}\frac{{{\rm{33}}}}{{{\rm{80}}}}{\rm{ = 0}}{\rm{.4125 = 41}}{\rm{.25\% }}\)

Step-by-step solution

Definition of product rule and also types of component

The product rule of probability refers to the occurrence of two or more independent occurrences at the same time. This is the sum of the likelihood of each of these occurrences occurring independently.

Determining the ways in which one can component of each type be selected 

Receiver, CD player, speakers, and turntable are the four sorts of components. With the exception of the receiver (which has five manufacturers) and speakers, all have four possible manufacturers (which have \({\rm{3}}\) possible manufacturers).

Receiver: \({\rm{5}}\) options

There are four methods to use a compact disc player.

There are three options for speakers.

\({\rm{4}}\)different ways to turn the turntable

The basic counting principle is as follows: If one event may happen in \({\rm{m}}\) ways and another can happen in \({\rm{n}}\)ways, the total number of ways the two events can happen in order is \({\rm{m*n}}\)

Make use of the basic counting principle

\({\rm{5*4*3*4 = 240}}\)

Determining the ways in which can components be selected if both the receiver and the compact disc player are to be Sony 

Counting principle: If one event may happen in\({\rm{m}}\)ways and another can happen in\({\rm{n}}\)ways, the number of ways the two events can happen in order is\({\rm{m*n}}\)

There is just one method to use the receiver (Sony), and there is only one way to use the compact disc player (Sony) (Sony)

\({\rm{1*1*3*4 = 12}}\)

Determining the ways in which components be selected if none is to be Sony?

The counting principle is as follows: If one event may happen in \({\rm{m}}\) ways and another can happen in \({\rm{n}}\) ways, the total number of ways the two events can happen in order is \({\rm{m*n}}\)

We are unable to choose Sony. The receiver only has four options (rather than five), the CD player only has three options (rather than four), and the turntable only has three options (instead of\({\rm{4}}\)).

\({\rm{4*3*3*3 = 108}}\)

Determining the ways in which selection be made if at least one Sony component is to be included

We know there are a total of 240 potential methods from section (a).

We know there are 108 methods to do it without using a Sony component because of section (c).

The total number of potential ways to choose at least one Sony component is then reduced by the number of conceivable ways to choose no Sony component:

\({\rm{240 - 108 = 132}}\)

Determining the probability that the system selected contains at least one Sony component

If we choose a Sony component, the other components can't have Sony components in them. Then, using the basic counting concept, we get:

Additional from the Sony receiver, there are no other Sony components: \({\rm{1*3*3*3 = 27}}\)

There are no additional Sony components save the compact disc player: \({\rm{4*1*3*3 = 36}}\)

There are no additional Sony components save the turntable: \({\rm{4*3*3*1 = 36}}\)

The number of positive outcomes divided by the total number of potential outcomes equals the probability:

\(\begin{aligned}{{\rm{P(\;At least one Sony component\;) = }}\frac{{{\rm{\# \;of favorable outcomes\;}}}}{{{\rm{\# \;of possible outcomes\;}}}}{\rm{ = }}\frac{{{\rm{132}}}}{{{\rm{240}}}}{\rm{ = }}\frac{{{\rm{11}}}}{{{\rm{20}}}}{\rm{ = 0}}{\rm{.55 = 55\% }}}\\{{\rm{P(\;Exactly one Sony component\;) = }}\frac{{{\rm{\# \;of favorable outcomes\;}}}}{{{\rm{\# \;of possible outcomes\;}}}}{\rm{ = }}\frac{{{\rm{27 + 36 + 36}}}}{{{\rm{240}}}}{\rm{ = }}\frac{{{\rm{99}}}}{{{\rm{240}}}}{\rm{ = }}\frac{{{\rm{33}}}}{{{\rm{80}}}}{\rm{ = 0}}{\rm{.4125 = 41}}{\rm{.25\% }}}\end{aligned}\)