Simulation Using Random Numbers

We can simulate the rolling a pair of dice by calling roll-a-die twice -- perhaps making a list out of the two results:

   (define pair-of-dice
       (lambda ()
           (list (roll-a-die) (roll-a-die))
       )
   )

Exercise 1

Check that pair-of-dice works correctly.

Exercise 2

Combine the ideas of procedures roll and pair-of-dice to obtain a procedure roll-dice that shows the results of throwing a pair of dice n times.

Exercise 3

Modify pair-of-dice to obtain a new procedure sum-of-dice that returns the sum of two dice. (As with pair-of-dice, the sum-of-dice procedure should not require any parameters.)

Exercise 4

Modify roll to obtain a procedure count-7s, which counts the number of 7's rolled when a pair of dice is thrown n times.

Tossing Coins

Previous laboratory work also showed how to simulate the tossing of a coin by generating real numbers between 0 and 1 and then considering a "head" to correspond to a number less than 0.5.

Exercise 5

Build on your previous work by writing a procedure tosses which simulates the flipping of a coin n times. The results of each toss should be placed on a list.

Exercise 6

Write a procedure number-of-heads that counts the number of heads that are obtained in n tosses.

Challenge Problem 1:

Consider the problem of rolling a pair of dice n times and counting the number of times that either a 7 or an 11 comes up.

What is wrong with the following procedure to accomplish this task?

   (define seven-or-11
       (lambda (n)
           (cond  ((<= n 0)  0)
                  ((or (= (sum-of-dice) 7) (= (sum-of-dice) 11))
                        (+ 1 (seven-or-11 (- n 1))))
                  (else (seven-or-11 (- n 1)))
           )
       )
   )

Write a correct procedure to solve this problem.

Challenge Problem 2:

Write a procedure double-heads which tosses a coin n times and determines whether a head ever comes up twice in a row.
(Hint: You might try adding another parameter which indicates if the previous toss were a head.)

Challenge Problem 3:

Modify the previous procedure to record the number of times a double head is obtained when a coin is tossed n times. (In your counting, you should consider three heads in a row as two double heads.)

Challenge Problem 4:

Modify your procedure from Challenge Problem 2, so that it continues the simulation until a double head appears and then prints the number of flips that were required to get the double head.

This document is available on the World Wide Web as

http://www.math.grin.edu/~walker/courses/151.sp04/readings/simple-simulations.xhtml

created February 5, 1997 last revised January 16, 2004
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.