Download Families of Four Activity/Project

Probability Project
You may work with up to three partners for this activity. You will turn in one copy of the write-up for the
whole group. No more than two groups per topic because you will be presenting these to the class in a four
to six minute presentation that includes a visual aid.
Your project should be typed, double-spaced, with pages numbered. It should have a title page listing the
names of all group members. All graphs and tables should be typed or neatly written in ink and drawn with
a straight edge. They should be included within the body of your paper at the appropriate point or referred
to by page and “figure” number.
Use good English skills—complete sentences with correct grammar, spelling and punctuation. Your writeup should contain the following paragraphs (paragraphs contain several sentences and are not numbered):
1.
2.
3.
4.
5.
6.
Introduction
Theoretical data
Description of simulation your group chose
Simulated and/or real data
Comparison of theoretical and simulated data
Conclusion.
A. Families of Four:
1. You need an introduction to your write-up, briefly describing the purpose of the entire activity.
2. For families of four children, assume that births of females and males are equally likely to occur. Find
the theoretical probability of families having 0, 1, 2, 3, and 4 girls, respectively. Show and/or describe
completely how you got your theoretical values. Summarize the results in the form of a probability
distribution table and a probability histogram. Calculate the mean () and standard deviation () for the
numbers of girls in theoretical families with four children. This theoretical data is the “standard” to use
to judge your sample (simulated and/or “real”).
3 and 4. Collect data from simulated families—you need (16 x number in your group) trials. Describe the
method you used completely. Find the number of girls in each of the 16 “families”. List your data then
summarize the results in a frequency table and frequency histogram. Discuss what you found. Suggested
simulations: (a) refer to a telephone book (record the last four digits only), (b) a random number table,
(c) your calculator or computer software. You can let the digits 0, 1, 2, 3, 4 represent girls and 5, 6, 7, 8,
9 represent boys or use odd/even for boys/girls. You may do the simulation by (d) coin tossing, (e) dice
rolling or any other appropriate random method. You may even include real families with four children.
5. Use a back-to-back stem-and-leaf plot, dot-plot or other graph to compare the shape, center and spread of
the theoretical and simulated and/or real data distributions. Find the actual mean and standard deviation
of the number of girls in the simulated and/or real families (sample data values) and compare them to the
values for the theoretical families (population values).
6. Write a conclusion that summarizes everything you discovered in this activity. It should refer to both
kinds of data, the similarities and differences between your results using each method and possible
reasons for any differences. Talk about how your data collection method might be a source of bias and
any effects of sample sizes.
You may also choose any of the following problems to work and write up in a similar manner to “Families
of Four”. If you are asked to perform a simulation you must describe your simulation and record each trial.
For example, if you were playing a game where you were rolling two dice to get double sixes, you might
state one trial like this: 6-4, 2-1, 3-6, 4-2, 6-6—5-rolls needed. I need to see exactly what you have done.
You may choose to actually "play" some of the games/problems instead of simulating them. Include these
real outcomes of each "trial" as you would a simulation.
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For each of these problems, you need to perform your simulation for (20 x number in group) trials. Just
coming up with the "correct" final answer is not sufficient. Presentation, documentation, and well-explained
conclusions are crucial. Remember, on all of these problems, you need to find the theoretical probability
(with work shown) and use it as the standard to which you compare your simulated data.
B. Cereal Toys: A certain brand of cereal is running
a promotion where you can get 1 toy in each box
purchased. There are 5 different toys. I wish to
collect one set of these toys to give as a Christmas
gift. Assuming the toys are randomly distributed
among the boxes, how many boxes should I expect
to buy before I have all 5 toys (theoretical)? What is
the average number of boxes you needed to buy
before a set was obtained (simulated)?
C. Dice Rolling: Italian gamblers used to bet on the total number of dots rolled
on three six-sided dice. They believed the chance of rolling a nine ought to equal
the chance of rolling a total of 10 since there were an equal number of different
ways to get each sum. However, experience showed that these did not occur
equally often. The gamblers asked Galileo for help with the apparent
contradiction, and he resolved the paradox. Can you do the same? Be sure to
explain both why the gamblers were confused and what the actual probabilities
are. Include evidence from actual trials or simulations and theoretical
calculations.
D. Chuck-A-Luck: The game Chuck-A-Luck is played as follows: A player bets
$M on an integer chosen from 1- 6 and then rolls three dice. If the number appears
on exactly one die, then the player wins the amount bet. The player wins twice the
amount bet if the number appears on two dice and three times the amount bet if the
number appears on all three dice. If the number does not appear at all, then the
player loses the $M. What is the probability of winning at each level? At any
level? For every n such bets of a fixed amount $M, what would be the House's
average net gain? (In other words, find  such that  = average gain by the house.)
E. Cut the Cards: Suppose we take a shuffled deck of 52 cards and make three cuts into the deck, where
each successive cut is deeper than the previous. What is the probability p of getting an Ace, a Deuce, or a
Jack with these three cuts?
F. Dice Rolling #2: [In 17th century France, the Chevalier de Mere posed a famous problem which led to
the development of probability theory. It was first solved by Blaise Pascal in collaboration with Pierre de
Fermat.] Determine which of the following two situations will give you better odds (if either). Determine
the solution mathematically and then perform a simulation. Do half of your trials for each
"game." (1) If you roll a fair die over and over until a Six is rolled, what is the
probability of rolling the Six within 4 rolls? (2) If one rolls two dice, what is the
probability of rolling a Double Six within 24 rolls?
G. Your Own Game: Design your own “game” similar to own shown here, get the teacher’s approval, and
continue as above—get simulated data and compare it to the theoretical probability for your game.
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