Chapter 5, Problem 1PP

(Probability) The probability that a phone call will last less than t minutes can be pproximated by the exponential probability function:

Probability that a call lasts less than t $minutes=1-e{}^{-t/a}$

$\begin{array}{l}aistheaveragecalllength.\hfill \\ eisEuler’snumber\left(2.71828\right).\hfill \end{array}$

For example, assuming the average call length is 2 minutes, the probability that a call will last

less than 1 minute is calculated as $1-e-1/2=\mathrm{0.3297.}$ Using this probability equation, write a

C++ program that calculates and displays a list of probabilities of a call lasting less than 1 minute to less than 10 minutes, in 1-minute increments.

Program Plan Intro

Program plan:

• Include the header files and using the namespace for standard I/O.
• Define the main function.
• Declare the variables x and z of integer type and y of double data type.
• Declare the essential variables.
• Use the loop to iterate.
• Calculate the probability that the call last less than t minutes.
• Display the list of probabilities in tabular format

Program description:

The main purpose of the program is to calculate theprobability of that a call will last less than one minute. This is calculated by the below formula: -

  $\begin{array}{l}p=1-e^{\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}\\ \\ \forall ei\text{s}\text{the}\text{Euler's}\text{number}(2.71828)\\ a\text{is}\text{the}\text{average}\text{call}\text{length}\\ t\text{is}\text{the}\text{minutes}\end{array}$

Explanation of Solution

Program:

//Importing the essential headers
#include <iostream>
#include <iomanip>
#include <math.h>
//Using the namespace for standard I/O
usingnamespacestd;
//Defining the main function
intmain()
{
//Declaring the essential variables 
double p;//to store the Probability
double e =2.71828;//to store the Euler’s number
int a=2;//to store the average call length
cout<<"Call Time\tProbability\n";
//Using the loop to iterate 
for(int t =1; t <10; t++)
{
//Calculating the probability that the call last less than t minutes
        p =1-pow(e,(-1*t)/(a));
//Displaying the list of probabilities in tabular format 
cout<<setprecision(0)<< fixed << t <<"\t\t"<<setprecision(4)<< p <<endl;
}
}

Sample output: