91Ó°ÊÓ

The term "probability" simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes—how likely they are—when we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

a)

Product Rule for k-Tuples

If we assume that we may pick the first element of an ordered pair in \({{\rm{n}}_{\rm{1}}}\)ways and the second element in \({{\rm{n}}_{\rm{2}}}\)ways for each selected element, then the number of pairings is \({{\rm{n}}_{\rm{1}}}{{\rm{n}}_{\rm{2}}}\)

Similarly, there are \({{\rm{n}}_{\rm{1}}}{{\rm{n}}_{\rm{2}}}{\rm{* \ldots *}}{{\rm{n}}_{\rm{k}}}\)potential\({\rm{k}}\)-tuples given an ordered collection of \({\rm{k}}\)components, where the \({{\rm{k}}^{{\rm{th }}}}\)can be selected in \({{\rm{n}}_{\rm{k}}}\)ways.

There are \({\rm{14}}\) lines and \({\rm{10}}\) pages in all. We can choose the first line of a sonnet in ten different ways, so \({{\rm{n}}_{\rm{1}}}{\rm{ = 10}}\); similarly, we can choose all of the other lines in ten different ways, so \({{\rm{n}}_{\rm{i}}}{\rm{ = 10,i = 1,2, \ldots ,14}}\). As a result,

\({{\rm{n}}_{\rm{1}}}{\rm{*}}{{\rm{n}}_{\rm{2}}}{\rm{* \ldots *}}{{\rm{n}}_{{\rm{14}}}}{\rm{ = 10*10 \ldots *10 = 1}}{{\rm{0}}^{{\rm{14}}}}{\rm{.}}\)

It is possible to write sonnets for \({\rm{10}}\). The first ten sonnets are included in this total.

b)

Product Rule for k-Tuples

If we assume that we may pick the first element of an ordered pair in \({{\rm{n}}_{\rm{1}}}\)ways and the second element in \({{\rm{n}}_{\rm{2}}}\)ways for each selected element, then the number of pairings is \({{\rm{n}}_{\rm{1}}}{{\rm{n}}_{\rm{2}}}\)

Similarly, there are \({{\rm{n}}_{\rm{1}}}{{\rm{n}}_{\rm{2}}}{\rm{* \ldots *}}{{\rm{n}}_{\rm{k}}}\)potential\({\rm{k}}\)-tuples given an ordered collection of \({\rm{k}}\)components, where the \({{\rm{k}}^{{\rm{th }}}}\)can be selected in \({{\rm{n}}_{\rm{k}}}\)ways.

We have \({\rm{14}}\) lines, and \({\rm{10}}\) pages. Imagine that, we delete first and last sonnet, now we have \({\rm{14}}\) lines and \({\rm{8}}\) pages. We can select first. line of a sonnet in \({\rm{8}}\) different. ways, therefore\({{\rm{n}}_{\rm{1}}}{\rm{ = 8}}\), similarly for all other lines, and they are \({\rm{14}}\) in total, we have\({{\rm{n}}_{\rm{i}}}{\rm{ = 8,i = 1,2, \ldots }}{\rm{.11}}\). Therefore

\({{\rm{n}}_{\rm{1}}}{\rm{*}}{{\rm{n}}_{\rm{2}}}{\rm{* \ldots *}}{{\rm{n}}_{{\rm{14}}}}{\rm{ = 8*8 \ldots *8 = }}{{\rm{8}}^{{\rm{14}}}}{\rm{.}}\)

sonnets can be created, which is omr number of favorable outcomes in

\({\rm{P(A) = }}\frac{{{\rm{\# of favorable outcomes in A}}}}{{{\rm{\# of outcomes in the sample space }}}}{\rm{.}}\)

The number of outcomes in the sample space is calculated in (a) and it is\({\rm{1}}{{\rm{0}}^{{\rm{14}}}}\). Therefore,

\({\rm{P(A) = }}\frac{{{{\rm{8}}^{{\rm{14}}}}}}{{{\rm{1}}{{\rm{0}}^{{\rm{14}}}}}}{\rm{ = 0}}{\rm{.044}}\)