Question

Problem 9) Given a biased coin analogy, if a bit vector is 30% full, what are the odds that 5 bits randomly chosen all fall within the filled section. Simulate the results and compare to calculation.

Answer 1

Manual Calculation:

Probability of single bit fall in 30% area is 30/100

Probability of all 5 bit fall in 30% area is (30/100)^5=0.00243

By computer simulation the output is is 0.002483

Programming:

Since there is nothing mentioned in the question, I am using C++ programming language for code development.

#include<iostream>                    //Header file inclusion
#include<stdlib.h>                      //Header file inclusion
using namespace std;                //Namespace std definition

void calculate_problem()             //Calculate function with return type void
{
//We can assume that there are 100 bit in vector and 30 are filled as 30% is full
// we will choose 5 numbers/bits from the range 1 to 100.
//We need to find how many times all five bits or numbers are less that 30 for a very big number of trials
int i=0,hit=0;                               //Variable declaration
double prob=0;                           //Variable declaration
while(i<1000000)                        //Number of trials are 1000000
{
int r1=rand()%100+1;
int r2=rand()%100+1;
int r3=rand()%100+1;
int r4=rand()%100+1;
int r5=rand()%100+1;
if(r1 <=30 && r2 <=30 && r3<=30 && r4<=30 && r5<=30)     //Condition checking
{
hit++;                                                                                       //Post increment of hit
}
i++;                                                                                          //Post increment of variable i
}
prob=(double)hit/1000000;
cout<<"Probability of falling in 30% area is "<<prob;              //Printing the probability
}
int main()                                                    //Main function with return type integer
{
calculate_problem();                                                              //Calling the calculate function
return 1;                                                                                 //Returning the value
}

OUTPUT:

I have run the program in online C++ compiler and the screenshot is attached.

29 int r4 rand()%100+1 39 int r5-and ( )%100+1; 31 if(r1 <30 && r2 <-30 && r3<-30 && r4-30 && r5-30) //Condition checking 32 33 hit++; 34 35 i++ 36 37 prob-(double)hit/1000000 38 cout << "Probability of falling in 38% area is "<<prob; 39 40 int main() 41 42 calculate_problem(); 43 return 1; //Post increment of hit //Post increment of variable //Printing the probability //Main function with return typ //calling the calculate function //Returning the value input Probability of falling in 30% area is 0.002483 . ..Program finished with exit code 1 Press ENTER to exit console.

Hope this helps.