Lab: Simulation

• Exercises
  • Exercise 0: Getting Started
  • Exercise 1: Testing random
  • Exercise 2: Rolling Dice
  • Exercise 3: Counting Dice
  • Exercise 4: Flipping Coins
  • Exercise 5: Searching for Paradise
  • Exercise 6: Sevens or Elevens
  • Exercise 7: ... or Doubles
  • Exercise 8: Repeated Heads
• Notes
  • Notes on Exercise 2

Exercises

Exercise 0: Getting Started

a. Scan through the reading on simulation.

b. Start DrScheme

Exercise 1: Testing random

a. Evaluate the expression (random 10) twenty times. What values do you get?

b. Try calling random a few times using 1 as a parameter. What values do you get?

c. Try calling random with -1 as a parameter. What value do you get?

d. Try calling random with other parameters. What effect does the parameter seem to have?

Exercise 2: Rolling Dice

a. Copy the roll-a-die and roll-dice procedures.

b. Using roll-dice, roll ten dice.

c. Using roll-dice, roll ten dice.

d. Did you get the same list of values each time?

e. What other procedures return different values each time you call them?

Exercise 3: Counting Dice

Write a procedure, (count-odd-rolls n) that counts the number of odd numbers that come up when rolling n six-sided dice.

Exercise 4: Flipping Coins

a. Write a procedure, (heads?) that simulates the flipping of a coin. Heads should return #t (which represents "the coin came up heads") half the time and #f (which represents "the coin came up tail") about half the time.

b. Write a procedure, (count-tails n) that simulates the flipping of n coins (using heads? to simulate each coin) and returns the number of times the coin is tails.

c. Use count-tails to test heads? by counting the number of heads you get in 1000 flips.

Exercise 5: Searching for Paradise

a. Write a procedure, (pair-a-dice), that simulates the rolling of two six-sided dice and prints out a pair of the results.

b. Write a procedure, (sum-a-dice), that simulates the rolling of two six-sided dice and then computes their sum.

c. Write a procedure, (count-sevens n) that simulates the rolling of n pairs of dice and counts the number of times the value 7 appears.

Exercise 6: Sevens or Elevens

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.

a. 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))))))

Hint: Try adding a display to sum-of-dice so that you can see how many times sum-of-dice is called.

b. Write a correct procedure to solve this problem.

Exercise 7: ... or Doubles

Extend your procedure from the previous exercise to count the number of times 7, 11, or ``doubles'' (two dice with the same value) come up in n rolls.

Exercise 8: Repeated Heads

a. Write a procedure, (double-heads? n), which tosses a coin n times and determines whether a head ever comes up twice in a row.

Do not make a list of flips and then scan through the list. However, you might try adding another parameter which indicates if the previous toss were a head.

b. Write a procedure, (count-double-heads n), which records 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.

Notes

Notes on Exercise 2

Just in case you don't have the reading handy, here's the code again.

;;; Procedure:
;;;   roll-a-die
;;; Parameters:
;;;   None
;;; Purpose:
;;;   To simulate the rolling of one six-sided die.
;;; Produces:
;;;   An integer between 1 and 6, inclusive.
;;; Preconditions:
;;;   None.
;;; Postconditions:
;;;   Returns an integer between 1 and 6, inclusive.
;;;   It should be difficult (or impossible) to predict which
;;;     number is produced.
(define roll-a-die
  (lambda ()
    (let ((tmp (random 6))) ; tmp is in the range [0 .. 5]
      (+ 1 tmp))))          ; result is in the range [1 .. 6]

;;; Procedure:
;;;   roll-dice
;;; Parameters:
;;;   n, an integer (the number of dice to roll)
;;; Purpose:
;;;   Roll n dice.
;;; Produces:
;;;   A list of integers, each between 1 and 6 (inclusive).
;;; Preconditions:
;;;   n >= 1.
;;; Postconditions:
;;;   Returns a list of length n.
;;;   Each element of the list is between 1 and 6 (inclusive).
;;;   The elements of the list are difficult (or impossible) to predict.
(define roll
  (lambda (n)
    ; If there are no dice left to roll, ...
    (if (<= n 0)
        ; then give an empty list of rolls
        null
        ; Otherwise, roll once and then roll n-1 more times.
        (cons (roll-a-die) (roll-dice (- n 1))))))