Question

A sonnet is a \({\rm{14}}\)-line poem in which certain rhyming patterns are followed. The writer Raymond Queneau published a book containing just \({\rm{10}}\) sonnets, each on a different page. However, these were structured such that other sonnets could be created as follows: the first line of a sonnet could come from the first line on any of the \({\rm{10}}\) pages, the second line could come from the second line on any of the \({\rm{10}}\) pages, and so on (successive lines were perforated for this purpose).

a. How many sonnets can be created from the \({\rm{10}}\) in the book?

b. If one of the sonnets counted in part (a) is selected at random, what is the probability that none of its lines came from either the first or the last sonnet in the book?

Short answer

a) There are \({\rm{1}}{{\rm{0}}^{{\rm{14}}}}\)sonnets can be created from the \({\rm{10}}\) in the book.

b) The probability is \({\rm{0}}{\rm{.044}}\).

Step-by-step solution

Definition

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.

Calculating the sonnets

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.

Calculating the probability

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,

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