91Ó°ÊÓ

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.

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

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

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)

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

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\({\rm{4}}\)).

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

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:

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

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:

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