Back to the Table of Contents

Statistical Probabilities and Distributions

Homework for Prob./Dist. Lesson 7

Note: Be sure to make clear your procedure for obtaining your random numbers. Specifically, be sure to provide enough information to repeat it EXACTLY!
  1. Enter the expression int(1+6rand) on your calculator then hit enter 5 times (or do randInt(1,6,5)). Use the first three "rolls" as red dice and the last two as the white. Calculate who won based on the rules printed in the lecture. (DO NOT ADD PIPS: Compare highest red with highest white---red wins only if larger. Compare second highest red with lowest white---white wins if equal or greater.) Repeat for a total of 12 battles. Tabulate your results here. If possible, pool your results with everyone else in the class and compare with the results cited in the lecture.

     

     

     

  2. Using a random number table (see pi below) or random digits on your calculator, generate 15 "families", stopping once each family has both a boy and a girl. Show your work and calculate the average number of children. Be sure to specify which method you used. Please specify 0->rand (where any number could be substituted for 0) so your results can be verified and/or repeated by someone else.

     

     

  3. Either link to a Monty Hall problem site and run some simulations there or conduct 25 trials as described in the lecture. The first 55 (since you are ignoring zeros) digits of pi are: 14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209.

     

     

     

  4. Large values of n! are occasionally needed. It may be impractical or too time consuming to calculate them by direct multiplication. A typical example might be when the Bureau of Fisheries asked Bell Labs for help finding the shortest route for getting samples from 300 locations in the Gulf of Mexico. There are 300! different possible routes. This is also known as the travelling salesman problem. Calculuate values for 50!, 100!, 200!, and 300! as described below and compare with offical results. (Give percentage error: (Observed-Expected)/Expected×100%.
    n!=10K where K=(n+0.5)log10n - 0.43429448n + 0.39908993.
    Note that this is Stirling's approximation (log n! (n+½)log n - n + ½log(2)) converted to log10. Be sure to use all the significant digits since three or four will be lost in the exponent.

     

     

     

    50! = 3.0414 x 1064
    100! = 9.33262 x 10157
    200! = 7.88658 x 10374
    300! = 3.06058 x 10614

     

  5. A student guesses answers to each of the 5 true/false questions on a quiz. Use the decimal expansion of pi and even/odd (don't care which is correct) to estimate the mean number of correct responses for 10 such students.

     

     

     

  6. Use the decimal expansion of pi (or a random number table/generator--specify which so your work can be reproduced) to estimate the average number of rolls of a single die necessary to get a 6. (Hint: Skip any outcomes that are not 1, 2, 3, 4, 5, or 6.)

     

     

     

  7. A ?...........................? of an experiment is a process that behaves the same way as the experiment, so that similar results are produced.

     

     

     

BACK TO LESSON ODD SOLUTIONS ACTIVITY CONTINUE