Chapter 9, Problem 27CR

To determine

Tocalculate:The probability thata diagonal chosen at random will be one of the shortest diagonal.

Answer to Problem 27CR

The probability is $\frac{2}{7}$ .

Explanation of Solution

Given information:

A regular decagon with 35 diagonals.

Formula used:

Diagonals of a decagon, $d=\frac{n(n-3)}{2}$

n = number of sides of a decagon.

Probability of event to happen is

  $P\left(E\right)=\frac{Numberoffavourableoutcomes}{TotalNumberofoutcomes}$

Calculation:

  

Decagon has 10 sides, i.e. n =10.

Let,

  $\begin{array}{l}d=\frac{n(n-3)}{2}\\ d=\frac{10(10-3)}{2}\\ d=\frac{10(7)}{2}\\ d=35\end{array}$

Probability means possibility.

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourableoutcomes and the total number of outcomes.

The shortest diagonal is one which skips only one vertex. There are 10 of these.

In our example, there are 10 favourableoutcomes.

There are 35 total number of outcomes, $i.e.d=35$

  $P\left(Oneoftheshortestdiagonal\right)=\frac{10}{35}=\frac{2}{7}=0.2857=28.57\%$