Exploring Probability
Name_____________________________
Casino and other games are based on probabilities. One such game, Craps, pays out cash awards when the player rolls a seven with two dice. But why a seven? Why not an eight? Or a twelve?
You will sample some of the possible combinations of rolling two dice and record your results. You will then try to predict what you would expect to roll, based on your results. Then you compare the theoretical probabilities with your actual results.
Collecting Data
· Roll a pair of dice 50 times. Record the sum of both dice for each of your rolls in the table below.
|
Roll
Number |
Result |
Roll
Number |
Result |
|
1 |
|
26 |
|
|
2 |
|
27 |
|
|
3 |
|
28 |
|
|
4 |
|
29 |
|
|
5 |
|
30 |
|
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6 |
|
31 |
|
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7 |
|
32 |
|
|
8 |
|
33 |
|
|
9 |
|
34 |
|
|
10 |
|
35 |
|
|
11 |
|
36 |
|
|
12 |
|
37 |
|
|
13 |
|
38 |
|
|
14 |
|
39 |
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|
15 |
|
40 |
|
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16 |
|
41 |
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17 |
|
42 |
|
|
148 |
|
43 |
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|
19 |
|
44 |
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20 |
|
45 |
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21 |
|
46 |
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22 |
|
47 |
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23 |
|
48 |
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24 |
|
49 |
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25 |
|
50 |
|
Count the total number of occurrences of each possible sum. Record them in the following chart:
|
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
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Based upon your results, how many times would you expect to roll each of the following if you were to roll the dice an extended number of times? Record your estimations in the following chart for the given number of rolls.
|
No.
Rolls |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
|
100 |
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500 |
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1000 |
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5000 |
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Explain how you came up with your predictions.
Complete the following table showing the sum of the dice by adding the appropriate rows and columns (similar to the multiplication table).
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|
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
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2 |
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3 |
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4 |
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5 |
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6 |
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This table shows all of possible combinations of rolling two dice.
1. How many possible ways are there to roll a 1? 2? 3? . . . 12? How can you tell?
These are called EVENTS
2. How many total possibilities are there?
This is called the SAMPLE SPACE
3. Do you notice any patterns in the chart? If yes, what are they? Why do you think this pattern exists? What do you think it means?
·
The Probability, P,
of an event is defined as the number of occurrences of an EVENT,
E, divided by the total number of possibilities in the SAMPLE
SPACE, S.
For example: The probability of rolling a one is: P(1) = 0 / 36 = 0. So there is zero chance of rolling a one with two dice.
Calculate the probabilities of rolling each of the possible sums in the same manner:
|
|
Results from original trial of 50 |
Guesses from 100 |
Guesses from 500 |
Guesses from 1000 |
Guesses from 5000 |
Theoretical Probability |
|
P(2) |
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P(2) |
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P(3) |
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P(4) |
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P(5) |
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P(6) |
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P(7) |
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P(8) |
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P(9) |
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P(10) |
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P(11) |
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P(12) |
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How do your results compare with the actual theoretical probability? How close were you? Why do you think you were so close or so off?
What are some things that can be done when rolling the dice that would help increase the chance of rolling a sum the exact number of times that we expect?