Simulation Using Random Numbers

Previous laboratory work simulated the rolling of a die, by defining a procedure roll-a-die using Chez Scheme's random procedure to obtain random integers 1, 2, 3, 4, 5, and 6. That work also simulated the rollng of a die several times, showing the results of each of n tosses. The resulting procedure was called roll.

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))
       )
   )

1. Check that pair-of-dice works correctly.

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.

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.)

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.

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.

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

• 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.

Work to be turned in:

• Answers for steps 2-6.
• A solution to one of the Challenge Problems (your choice of which).

This document is available on the World Wide Web as

http://www.math.grin.edu/~walker/courses/151.fa00/lab-simulation.html