Chapter 6.4, Problem 3BE

To determine

To enter the program and run it for several times for large values of D like $100,400,800$ and also find whether the probability of broken stick forming a triangle is less than or greater than $\frac{1}{2}$ .

Explanation of Solution

Given:

Break a stick into three pieces to get the probability that you can join the pieces end-to-end to form a triangle. If the sum of the length of any two pieces is less than or equal to that of the third, a triangle can’t be form. This is known as Triangle Inequality. By an experiment your class can estimate the probability that three pieces of broken stick will form a triangle.

Calculation:

Let D, N, I, X, Y, R, S and T variables used in program. Where D stands for number of sticks you have to break, N stands for first end of stick that is 0, I stands for variable of for loop assign from 1 to D, X and Y are stand for the length of points distance from initial point that is 0, R assigned as X, S assigned as Y − R and T assigned for 1 − R − S. Consider the program below

Program:

'Print this statement "SIMULATION-BREAKING STICKS TO MAKE TRIANGLES" on output screen.

10 PRINT "SIMULATION-BREAKING STICKS TO MAKE TRIANGLES"

'PRINT use to create blank line.

20 PRINT

30 PRINT "HOW MANY STICKS DO YOU WANT TO BREAK”;

40 INPUT D

50 LET N=0

60 FOR I=1 to D

70 LET X=RND(1)

80 LET Y=RND(1)

90 IF X>=Y THEN 70

100 LET R=X

110 LET S=Y.R

120 LET T=1.R.S

130 IF R+S<=THEN 170

140 IF S+T<=R THEN 170

150 IF R+T<=S THEN 170

160 LET N=N+1

170 NEXT I

180 LET P =N/D

190 PRINT

200 PRINT “THE EXPERIMENTAL PROBABILITY THAT”

210 PRINT “A BROKEN STICK CAN FORM A TRIANGLE IS”, P

220 END

Sample Output:

SIMULATION-BREAKING STICKS TO MAKE TRIANGLES

HOW MANY STICKS DO YOU WANT TO BREAK ?100

THE EXPERIMENTAL PROBABILITY THAT

A BROKEN STICKS CAN FORM A TRIANGLE IS 0.28

Output Explanation:

SIMULATION-BREAKING STICKS TO MAKE TRIANGLES

Enter number of sticks as 100 which you want to break that is value of variable D

HOW MANY STICKS DO YOU WANT TO BREAK ?100

THE EXPERIMENTAL PROBABILITY THAT

Than you get experimental probability is equal to 0.28

A BROKEN STICKS CAN FORM A TRIANGLE IS 0.28

The probability is less than $\frac{1}{2}$

SIMULATION-BREAKING STICKS TO MAKE TRIANGLES

HOW MANY STICKS DO YOU WANT TO BREAK ?400

THE EXPERIMENTAL PROBABILITY THAT

A BROKEN STICKS CAN FORM A TRIANGLE IS 0.2275

Output Explanation:

SIMULATION-BREAKING STICKS TO MAKE TRIANGLES

Enter number of sticks as 400 which you want to break that is value of variable D

HOW MANY STICKS DO YOU WANT TO BREAK ?400

THE EXPERIMENTAL PROBABILITY THAT

Than you get experimental probability is equal to 0.2275

A BROKEN STICKS CAN FORM A TRIANGLE IS 0.2275

The probability is less than $\frac{1}{2}$

SIMULATION-BREAKING STICKS TO MAKE TRIANGLES

HOW MANY STICKS DO YOU WANT TO BREAK ?800

THE EXPERIMENTAL PROBABILITY THAT

A BROKEN STICKS CAN FORM A TRIANGLE IS 0.26375

Output Explanation:

SIMULATION-BREAKING STICKS TO MAKE TRIANGLES

Enter number of sticks as 800 which you want to break that is value of variable D

HOW MANY STICKS DO YOU WANT TO BREAK ?800

THE EXPERIMENTAL PROBABILITY THAT

Than you get experimental probability is equal to 0.26375

A BROKEN STICKS CAN FORM A TRIANGLE IS 0.26375

The probability is less than $\frac{1}{2}$