Cory Butts
Statistics
Coin Toss Project
Professor Fricke
October 30, 2014
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Introduction
There are many purposes to this coin toss experiment. The first purpose is for the students
to be able to apply probability concepts into real life business situations. Secondly, the students
are expected to be able to compare the various discrete probability distributions. Third, the
students differentiated between computational, subjective, and empirical approaches to
probability. In order to do so, the students had to identify the sample space, outcomes, and events
of the coin toss experiment. The fourth purpose is to compute the probabilities of simple real life
events. Lastly, the students became familiar with the use of technology for summarizing and
visualizing the probability. In order for the students to obtain the goals above a specific
procedure had to be followed.
The first step in the coin toss experiment was to estimate the subjective probability using
a cup full of ten fair coins. The estimates referred to the percent of time that each number of
heads could turn up per 10 coin toss. After the guesses were made, a computer generated coin
toss program tossed the coins 60 times for a total of 600 coins being tossed. The next step was to
copy the results into excel calculate the cumulative number of heads and cumulative percent
heads. Next, we used the data to compute the empirical and computational probability. Using
these results, the next steps were to construct a line graph of the cumulative percent heads and a
bar graph showing the empirical vs computational probabilities.
Results
1. What percent of the time do you expect to get 5 heads? .130
2. What percent of the time do you expect to get 3 heads? .115
3. What percent of the time do you expect to get 7 heads? .095
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4. What percent of the time do you expect to get no heads? .035
5. What percent of the time do you expect to get all heads? .045
6. Use your guesses above to fill in the chart below for all of the outcomes.
X = Heads
0
1
2
3
4
5
6
7
8
Prob(X)
.035
.125
.075
.115
.095
.130
.085
.095
7. What is the expected value for heads after tossing 600 coins?
.110
9
10
.090
.045
4.965
8. How is the probability of getting three heads (# 2 above) related to the probability of
getting three tails (# 3 above)? Explain your answer.
The probabilities of each happening have an equal chance because it is a binomial
distribution.
9. How is the probability of getting three heads related to the probability of getting seven
tails? Explain your answer.
In the instance of getting three heads it means that you will get seven tails because they have
to equal 1.
Below, there are two different graphs. This first graph represents a line graph of the
cumulative percent heads. The x axis has the number of tosses and the y axis has the
cumulative probability ranging from 0.4-0.6. As you can see, during the beginning tosses, the
cumulative probabilities fluctuate the most. Typically, the more tosses thrown, the closer that
the cumulative number of heads reaches fifty percent.
The second graph is a bar graph that represents the empirical and computational
probability of number of heads per 10 coins tossed. This computational bar graph is
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symmetrical. The empirical is skewed right. Both graphs are unimodal. Columns 1,2,9, and
10 are all unusual events.
Cumulative Probability
Cumulative Percent Heads
0.600
0.580
0.560
0.540
0.520
0.500
0.480
0.460
0.440
0.420
0.400
Cumulative Percent
Heads
1 6 11162126313641465156616671
Toss Number
Empirical vs. Computational
0.300
Probability
0.250
0.200
0.150
Empirical Probability
0.100
Computational Probability
0.050
0.000
1 2 3 4 5 6 7 8 9 10 11
Number of Heads
Conclusion
The results of this project are show in the line graph, bar graph, and the data from the
excel spreadsheet. With that said, the results demonstrate that there are many possibilities that
can occur when flipping 10 coins. Theoretically, by flipping 10 coins, most of the time 5 heads
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and 5 tails should be obtained. The data reflects this to be true most of the time. As one can see,
most of the data hovers towards the mean. The further away from the mean, the less likely an
event (number of heads). This experiment also shows that it is unusual for all or 9/10 coins are
heads and likewise with none or one of the coins to be heads. Overall, the line graph and bar
graph do an excellent job of illustrating the entirety of the project’s results.
Appendix A
This particular project taught me how to use statistics in possible real life situations.
With that said, this project prepared me how to use Microsoft Excel which is a program that is
used by many companies in the United States. Also, this project taught me how to incorporate
statistical information into an excel spreadsheet and use that information to make multiple
graphs. Meaning, I was able to take the heads and tails probabilities and use it to compare
discrete probability distributions. Also, I learned how to use the data obtained from the virtual
coin toss to distinguish between the computational, empirical, and subjective approaches to
probability. All in all, this project has certainly prepared me for future job assignments that
involve statistical information and using excel spreadsheets. I look forward to using this
knowledge to my advantage when searching for future jobs that require these skills.
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Appendix B: Charts 1 and 2
Toss Number Number of Heads Cumulative Heads Total Coins Tossed Cumulative Percent Heads
4
1
4
10
0.400
3
2
7
20
0.350
7
3
14
30
0.467
7
4
21
40
0.525
4
5
25
50
0.500
2
6
27
60
0.450
5
7
32
70
0.457
7
8
39
80
0.488
6
9
45
90
0.500
4
10
49
100
0.490
4
11
53
110
0.482
2
12
55
120
0.458
13
5
60
130
0.462
4
14
64
140
0.457
4
15
68
150
0.453
3
16
71
160
0.444
5
17
76
170
0.447
4
18
80
180
0.444
4
19
84
190
0.442
3
20
87
200
0.435
7
21
94
210
0.448
9
22
103
220
0.468
8
23
111
230
0.483
6
24
117
240
0.488
4
25
121
250
0.484
3
26
124
260
0.477
27
6
130
270
0.481
3
28
133
280
0.475
4
29
137
290
0.472
7
30
144
300
0.480
3
31
147
310
0.474
8
32
155
320
0.484
5
33
160
330
0.485
3
34
163
340
0.479
2
35
165
350
0.471
5
36
170
360
0.472
5
37
175
370
0.473
5
38
180
380
0.474
4
39
184
390
0.472
6
40
190
400
0.475
7
41
197
410
0.480
5
42
202
420
0.481
7
43
209
430
0.486
6
44
215
440
0.489
5
45
220
450
0.489
4
46
224
460
0.487
6
47
230
470
0.489
8
48
238
480
0.496
6
49
244
490
0.498
6
50
250
500
0.500
5
51
255
510
0.500
5
52
260
520
0.500
3
53
263
530
0.496
6
54
269
540
0.498
7
55
276
550
0.502
6
56
282
560
0.504
7
57
289
570
0.507
4
58
293
580
0.505
6
59
299
590
0.507
60
2
301
600
0.502
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X= No. of Successes Guess Probability of Successes Actual Number of Times Empirical Probability Computational Probability
0
0.035
0
0.000
0.000976563
1
0.125
0
0.000
0.009765625
2
0.075
4
0.067
0.043945313
3
0.115
8
0.133
0.1171875
4
0.095
13
0.217
0.205078125
5
0.13
11
0.183
0.24609375
6
0.085
11
0.183
0.205078125
7
0.095
9
0.150
0.1171875
8
0.11
3
0.050
0.043945313
9
0.09
1
0.017
0.009765625
10
0.045
0
0.000
0.000976563
Sums
1
60
1.000
1