Chapter 4, Problem 1R

Explanation of Solution

Preconditions assumed for solving the problem:

Consider the city with people who have a habit of enjoying the meal only if it is the best from the experiences in life. Otherwise, they do not accept the meal.

• Let us assume that the meal is distributed uniformly then the expected number of times the people are happy with their meals are given as follows:

Harmonic series:

It is the sum of reciprocals of first “n” Natural numbers. It is represented in the form as:

$\begin{array}{c}{H}_n=1+\frac{1}{2}+\frac{1}{3}+\mathrm{...}+\frac{1}{n}\\ ={\displaystyle \sum _{i=1}^N\frac{1}{i}}\end{array}$

Expected number of people happy with the meals:

To determine the number of people who are happy with their meals are estimated using the harmonic series.

Consider the harmonic series into given scenario, let us take $X_i$ as sequence of independent variables that is uniformly distributed over the interval (0, 1).

The quality of the meal of a person is represented using the variable $X_i$. So, the probability of the people happy at “$n$” number of meal is,

$P(X_n>X_i,\forall i<n)=\frac{1}{n}$

Thus, the expected value of people who has happy meal is as follows:

${\displaystyle \sum _{n=1}^N\frac{1}{n}}$

Where, “N” represents the count of number of meals in life.