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The people who first populated the Americas also invented and enjoyed playing games of chance. For example, the Mayan civilization that flourished for over three thousand years between 1500 B.C. to 1700 A.D. is remembered not only for its pyramids and temples, calendars, astronomy, and mathematics, but also for the music, dress, and games of its people. It is not surprising that this culture that depended so heavily on the cultivation of corn developed games of chance using kernels of corn as markers.
In this lesson you will see how probability has always been part of everyday life. You will play an ancient Mayan game of chance, collect data from samples, and use your results to make predictions about the populations from which you took the samples.
You will now explore the concept of fairness in games of chance.
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Games that are based on chance are part of almost every culture. The ancient Mayans played one such game called Bul (pronounced Bool). The Mayans didn't have dice, so they tossed four flat kernels of corn with a black mark burned on one side of each kernel. After tossing the corn, they counted the number of burned sides showing. The following table shows how counts of 1, 2, 3, 4, and 5 are scored.
| Toss |
|
|
|
|
|
| Count | 1 | 2 | 3 | 4 | 5 |
The game of Bul is still played today in Central American countries, such as Honduras. Each player uses five game pieces; the pieces could be pebbles, shells, dried beans, or seeds. The game "board" is made with 15 objects placed in a row or 15 marks etched in the ground. The 14 spaces between them is where the game pieces are moved.
Two players start at opposite ends of the board. The game begins with each player placing a game piece at his or her end. Each player gets two throws of corn in a row and moves the number of spaces indicated after each throw. Each player has only one game piece for each round of play.
When a game piece gets to the end of the board, it is re-entered at the end where it started, as if the board were circular. A round of play ends when one player captures an opponent's piece by landing on the space occupied by that piece. Another round then starts, with each player placing a new game piece at his or her end. This continues until one player wins the game by capturing all of the opponent's game pieces.
You can find out more about the game Bul on the Internet at the following website:
http://www.halfmoon.org/bul.html
1.
a.
Do you think Bul is a fair game? Explain.
b. How can you test your answer?
To test the fairness of Bul, you can play a modified version of the game. Here are the rules for a modified game of Bul:
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2. Play 20 rounds of the modified game, using the game board shown. Remember, blue tosses first to start every round. You may ask someone to play with you or you may play it alone. Record your results in a table like the one that follows.
Colour Who Won the Round? (tallies) Total Rounds Won Blue Red
3. Based on your results, do you think this is a fair game?
4. What is the fewest number of tosses needed to complete a round?
5. What is the greatest number of tosses that may be needed to complete a round?
6. Julio enjoyed playing modified Bul at home. When he showed the game to his friend, Breanne, she said she didn't think she would beat him very often. "You've played it so many times that you must be very good at it." What would you tell Breanne?
Now it's time for you to show your home instructor what you have been learning.
Play a game such as Snakes and Ladders, Go Fish, or Checkers with your home instructor.
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Discuss the importance of skill and chance in the game.
Now you will conduct a probability experiment with letters.
Have you ever played Scrabble®? Scrabble® is a popular crossword game in which players take turns spelling words with letter tiles. Did you ever wonder how the manufacturers of Scrabble® decided how many tiles to make for each letter of the alphabet?
1. Which five letters do you think are used most often in the English language?
2. Which letter do you think is used least often in the English language?
You can test your prediction by choosing a sample of writing in the English language, counting the number of times each letter is used, and then listing the letters in order of frequency.
A sample is a part of the population from which information is gathered. The population is the entire group of people or things for which information is needed.
3. Turn to page 187 in your textbook.
a. How many letters do you think there are in the yellow box on page 187?
b. Do you think this would be a suitable sample to use to check the predictions you made in questions 1 and 2? Explain.
c. Use a table like the following to tally the frequencies of all the letters inside the yellow box on page 187. (If possible, work with your home instructor or someone else who can read the letters to you as you record the tallies.)
Frequency of Letters on Page 187 of Textbook Total A B C D-W X Y Z Total
4.
a.
Write the letters you tallied in order from greatest frequency to
least frequency.
b. How do the resulting frequencies compare with your predictions?
5.
a.
How do you think the letter frequencies in a sample of English writing taken from a book you are
currently reading, or from a newspaper article, would compare with the letter frequencies in the textbook
sample?
b. Choose a sample of English writing with about the same number of letters as the textbook sample. Use a table like the one you used in question 3.c. to tally the letter frequencies for your chosen sample. (If possible, work with your home instructor or someone else who can read the letters to you as you record the tallies.)
6.
a.
Write the letters in your sample in order from greatest frequency to least frequency.
b. How do the resulting frequencies compare with the textbook sample?
7.
a.
Suppose you were making your own crossword board game that had 98 letter tiles. In a table like
the one below, show how many of the 98 tiles you would use for each letter. Then write the letters in order
from the greatest frequency to the least frequency.
Letter A B C D E F G H I J K L M Number
b. Scrabble uses 98 letter tiles (plus 2 blank tiles). Compare your tiles with the 98
Scrabble tiles. Explain how the two sets of tiles are the same and how they are different.
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You will now conduct a probability experiment with M&M's® candies.
If you enjoy M&M's® candies, you probably have a favourite colour. Is it blue or red? You may be curious about your chances of getting your favourite colour when you buy a package of M&M's®.
To investigate, you will need a 48-g package and a 159-g package of milk chocolate M&M's® as samples.
1. What is the population for which you are collecting data?
2. What colours are there for milk chocolate M&M's® candies?
3.
a.
Predict which colour you think will have the greatest frequency.
b. Predict which colour you think will have the least frequency.
4. Carefully empty your package of milk chocolate M&M's®, and use tables like the ones below to record your data.
Colours of
M&M's®Number of Each Colour in My 48-g
Sample (tallies)Total
FrequencyBlue Brown Green Orange Red Yellow
Colours of
M&M's®Number of Each Colour in My 159-g
Sample (tallies)Total
FrequencyBlue Brown Green Orange Red Yellow
5.
a.
Was your prediction in question 3.a. about the most frequent colour correct?
b. Was your prediction in question 3.b. about least frequent colour correct?
6. The following table indicates the fraction of each colour of milk chocolate M&M's® that the company makes. Use these decimal fractions to calculate the number of each colour expected.
Colours of
M&M's®Fraction of Each
ColourNumber Expected
in a 48-g Package
(with 60 M&M's®)Number Expected
in a 159-g Package
(with 180 M&M's®)Blue 0.1 Brown 0.3 Green 0.1 Orange 0.1 Red 0.2 Yellow 0.2
7. Compare the results for your samples with the completed table.
a. Explain how the results for your 48-g sample compare.
b. Explain how the results for your 159-g sample compare.
8. A larger sample usually predicts what a population is like better than a smaller sample does. Is this true for your two samples? If not, why do you think this is so.
9. Why doesn't every package of milk chocolate M&M's® in a particular size have the same number of each colour?
10. Some colours used for M&M's® have been changed over the years. In 1941, when M&M's® were first introduced, the colours were brown, red, yellow, green, orange, and violet. The violet colour was replaced first by tan in 1949, and again by blue in 1995.
Why do you think they made the colour changes, even though the candies all taste the same?
To discover some interesting facts about M&M's®, you may wish to visit the following website:
http://www.m-ms.com
1. In the following message, each letter of the alphabet has been replaced with a different letter. Find the message.
XUW XKGIJEW YZXU IFZDC P AGNW EZQW XUZF GDW ZF
XUPX ZX ZF WPFR LGK GXUWK TWGTEW XG ZL XUWR
QDGY PJGIX EWXXWK LKWBIWDAZWF.
2. In the following message, each letter of the alphabet has been replaced with a different symbol. Find the message.
The puzzles you have just completed are called cryptograms. If you would like to make a cryptogram of your own or solve another cryptogram, you will find the following website helpful:
http://www.wordles.com
In this lesson you saw how probability has always been part of everyday life. You played an ancient game of chance, collected data from samples, and used your results to make predictions about the populations from which you took the samples.
©2000-2001 www.arttoday.com
Games of chance have a long history in the Americas, from the ancient game of Bul to modern games, such as Monopoly®. Do you have a favourite game that involves the element of chance or luck? Do you think people will still be playing this game three thousand years from now?
1.
a.
It seems like it is a fair game because everything is based on chance.
b. I can test my answer by playing the game over and over to see if both players win about the same number of times.
Note: You may choose to repeatedly run the simulation of this Mayan game found at the following website:
http://www.halfmoon.org/bul.html
2. Sample data are provided.
Colour Who Won the Round? (tallies) Total Rounds Won Blue 
12 Red 
8
3. This seems to be a fair game because they both won about ten rounds. I suppose I could play it several more times and see if they both win about the same number of times.
4. It is possible to complete a round in as few as two tosses.
5. I suppose it is possible to keep tossing forever and never get a winner.
6. The outcome of this game is not influenced by skill or experience. The two players have an equally likely chance of winning.
Sample answers are given.
1. I think A, E, I, O, and R are the five letters used most often in the English language.
2. I think X is the letter used least often in the English language.
3.
a.
There are about 1000 letters on page 187.
b. I think this would be a suitable sample to use to check my predictions because the words appear to be everyday words, and there are lots of them.
c.
Frequency of Letters on Page 187 of Textbook Total A 
62 B 
15 C 
31 D 40 E
110 F 
16 G 25 H 
43 I 
36 J K 15 L 
42 M 
16 O
90 P 12 Q R 
49 S 
59 T
67 U 
47 V 5 W 30 X Y 
29 Z 
1 Total 905
4.
a.
The letters on page 187, from greatest frequency to least frequency, are E, O, T, A, N, S, R, U,
H, L, D, I, C, W, Y, G, M, F, B, K, P, V, and Z. The letters J, Q, and X were not in the sample.
b. I think my predictions were pretty good. For example, three of my predicted letters, A, E, and O, were the same as the sample. I also predicted I and R. R is the seventh letter in the sample and I is the twelfth letter in the sample.
5.
a.
I think the letter frequencies in a book I am reading or in a newspaper article would be similar
to the textbook sample.
b. As my sample, I used page 29 in the book Spirit in the Rainforest by Eric Wilson.
Frequency of Letters on Page 187 of Textbook Total A
74 B 
14 C 22 D 
48 E
104 F 
10 G 18 H 57 I 
63 J 1 K 22 L 38 M 18 O
72 P 13 Q 1 R 
52 S 50 T
87 U 32 V 17 W 19 X 1 Y 20 Z 1 Total 926
6.
a.
The letters in my sample, from greatest to least frequency, are E, T, A, O, N, I, H, R, S, D, L,
U, C, K, Y, W, G, M, V, B, P, F, J, Q, X, and Z.
b. The frequencies in my sample are similar to the textbook sample. The letter E was the most frequent letter in both samples. The next four most frequent letters in both samples were the same, although they were not in the same order. The letter Z appeared only once in each sample. The letters J, Q, and X were not in the textbook sample, and each of them appeared only once in my sample.
7.
a.
I would make the following letter tiles.
Letter E O T A N S R U H L D I C Number 12 8 7 7 6 6 6 5 5 5 5 5 3
W Y G M F B K P V Z J Q X 3 2 2 2 1 1 1 1 1 1 1 1 1
b. Scrabble® uses the following 98 tiles (plus 2 blank tiles).
Letter E A I O N R T D L S U G B Number 12 9 9 8 6 6 6 4 4 4 4 3 2
C F H M P V W Y J K Q X Z 2 2 2 2 2 2 2 2 1 1 1 1 1
My letter tiles are similar to the Scrabble® tiles. I had the same number of tiles as Scrabble® for 11 of the letters: E, O, R, D, Y, M, K, Z, J, Q, and X. Most of the others were pretty close. The two letters that differed the most were H and I. I had three more H's and four less I's than Scrabble®.
1. The population is all the milk chocolate M&M's® that are made.
2. Milk chocolate M&M's® are blue, brown, green, orange, red, and yellow.
3. A sample answer is given.
a. I think red will have the greatest frequency.
b. I think brown will have the least frequency.
4. A sample answer is given.
Colours of
M&M's®Number of Each Colour in My 48-g
Sample (tallies)Total
FrequencyBlue 7 Brown 
21 Green 4 Orange 7 Red 1 Yellow 19 Total 59
Colours of
M&M's®Number of Each Colour in My 159-g
Sample (tallies)Total
FrequencyBlue 20 Brown
53 Green 21 Orange 18 Red 33 Yellow 32 Total 177
5. A sample answer is given.
a. No, my prediction wasn't correct for either of my samples. The most frequent colour in both samples was brown.
b. No, my prediction wasn't correct for either of my samples. The least frequent colour in the 48-g sample was red. The least frequent colour in the 159-g sample was orange.
6.
Colours of
M&M's®Fraction of Each
ColourNumber Expected
in a 48-g Package
(with 60 M&M's®)Number Expected
in a 159-g Package
(with 180 M&M's®)Blue 0.1 6 18 Brown 0.3 18 54 Green 0.1 6 18 Orange 0.1 6 18 Red 0.2 12 36 Yellow 0.2 12 36
7. Sample answers are given.
a. In my 48-g sample, the numbers of blue, brown, green, and orange are close to the expected numbers. I only have one red compared to the expected number of 12 and 19 yellow compared to the expected number of 12.
b. In my 159-g sample, the numbers of all the colours are close to the expected numbers. There are 18 orange, which is exactly the expected number.
8. A sample answer is given.
My larger sample predicts the population better than my smaller sample. Most colours in both samples were close to expected numbers, but two colours in my 48-g sample were not close at all.
9. Every package of milk chocolate M&M's® in a particular size doesn't have the same number of each colour because the candies are mixed in very large amounts before the packages are filled.
10. Sample answers are given.
I think the company changed to colours that they thought their customers would like better. If people like the looks of the candies, maybe they will buy more.
1. THE TROUBLE WITH USING A CODE LIKE THIS ONE IS THAT IT IS EASY FOR OTHER PEOPLE TO SOLVE IF THEY KNOW ABOUT LETTER FREQUENCIES.
2. This is the last challenge problem. I hope you have enjoyed solving them.
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