Simulation
Simulation


Simulation imitation of chance
behavior based on a model that
accurately reflects the phenomenon
under consideration
By observing simulated outcomes,
researchers gain insight on the real
world.
Simulation

Why use simulation?
Some situations do not lend themselves to
precise mathematical treatment.
Others may be difficult, time-consuming,
or expensive to analyze. In these
situations, simulation may approximate
real-world results; yet, require less time,
effort, and/or money than other
approaches.
How to Conduct a
Simulation
A simulation is useful only if it closely mirrors
real-world outcomes. The steps required to
produce a useful simulation are below:
State the problem or describe the random
phenomenon.
2. State the assumptions.
3. Assign digits to represent outcomes (die, spinner, random
1.
digit table, calc.)
Simulate many repetitions
5. State your conclusions.
4.
(until outcome shows stable pattern)
Simulation Example
In this section, we work through an example
to show how to apply simulation methods to
probability problems.
Problem Description
On average, suppose a baseball player hits a
home run once in every 10 times at bat.
Using simulation, estimate the likelihood that
the player will hit 2 home runs in consecutive
at bats.
Solution
Use five steps required to produce a useful
Simulation:
1.
Problem:
–
2.
State Assumptions:
–
–
3.
at bats are independent of each other (that is, what happens during
one at bat does not influence the next at bat)
Homerun occurs 10% of at bats
Assign Digits:
–
–
–
–
4.
Estimate the likelihood that the player will hit 2 home runs in
consecutive at bats
two possible outcomes: home run or no home run
“2” represents a home run and other digits represent no home run
“22” represents back to back home runs
Any other two digit combination represents failure to hit back to back
homeruns
Simulate Many Repetitions:
– Stat Trek’s random digit generator was utilized for this example
– In this example, the list of random numbers consists of 500 2-digit
pairs
5.
State Conclusions
–
In the list, we found 6 occurrences of "22", which are highlighted in
red in the table
Random Numbers
42
86
36
17
30
74
51
32
45
89
39
46
59
04
53
34
74
92
68
49
62
06
82
63
76
99
92
90
20
82
75
73
03
64
31
56
86
47
14
39
69
30
97
22
52
13
80
05
35
96
02
76
28
38
80
27
06
89
31
78
57
41
83
30
08
18
34
81
59
70
85
32
11
21
36
65
24
88
84
11
43
08
03
31
90
54
49
77
09
73
69
96
29
20
15
16
75
39
35
62
04
53
34
74
92
68
49
62
06
82
63
76
21
70
78
91
09
85
66
79
23
99
80
93
38
14
39
69
30
97
22
52
13
80
05
35
96
63
88
17
79
46
44
00
35
28
27
83
18
11
30
08
18
34
81
59
70
85
32
11
21
36
10
89
72
14
61
40
24
65
12
91
01
17
64
09
73
69
96
29
20
15
16
75
39
35
62
95
85
81
82
41
05
01
28
61
25
20
48
07
70
78
91
09
85
66
79
23
99
80
93
38
87
95
08
26
02
83
96
40
16
98
10
55
04
88
17
79
46
44
00
35
28
27
83
18
11
10
40
01
94
93
22
84
77
75
05
67
60
58
89
72
14
61
40
24
65
12
91
01
17
64
42
53
68
15
94
04
19
93
45
55
97
44
23
85
81
82
41
05
01
28
61
25
20
48
07
71
67
94
26
90
86
14
73
37
32
33
92
56
95
08
26
02
83
96
40
16
98
10
55
04
12
25
43
19
00
13
57
33
15
27
72
21
29
40
01
94
93
22
84
77
75
05
67
60
58
88
50
43
41
71
33
26
24
54
16
09
07
37
53
68
15
94
04
19
93
45
55
97
44
23
06
48
95
74
84
00
47
25
36
51
98
77
87
67
94
26
90
86
14
73
37
32
33
92
56
52
79
12
02
98
99
58
22
18
45
78
42
37
25
43
19
00
13
57
33
15
27
72
21
29
42
86
36
17
30
74
51
32
45
89
39
46
59
50
43
41
71
33
26
24
54
16
09
07
37
99
92
90
20
82
75
73
03
64
31
56
86
47
48
95
74
84
00
47
25
36
51
98
77
87
02
76
28
38
80
27
06
89
31
78
57
41
83
79
12
02
98
99
58
22
18
45
78
42
37
65
24
88
84
11
43
08
03
31
90
54
49
77
This simulation predicts that the player will hit
consecutive home runs 6 times in 500 at bats.
Thus, the simulation suggests that there is a
1.2% chance that a randomly selected pair of at
bats would consist of two home runs.
The actual probability, based on the
multiplication rule, states that there is a 1.0%
chance of hitting consecutive home runs.
While the simulation is not exact, it is very close.
And, if we had generated a list with more random
numbers, it likely would have been even closer.
1.
Orders of frozen yogurt flavors have the following
relative frequencies: 38% chocolate, 42% vanilla,
20% strawberry.
State problem.
–
2.
State assumptions.
–
–
–
3.
5.
Frequencies are the same as stated
Assume that customers order one flavor only
Customers’ choices of flavors do not influence one
another
Assign digits.
–
–
–
4.
Simulate 10 frozen yogurt sales based on this recent
history.
00-37 chocolate
38-79 vanilla
80-99 strawberry
Simulate.
State your conclusions.
Simulate: Use Calculator to simulate
– Results: Run 20 numbers
State Conclusions:
– Chocolate:
– Vanilla:
– Strawberry:
A couple plans to have children until they have a girl or until
they have four children, whichever comes first.
Problem:
– Simulate and estimate probability that the strategy will
produce girl
Assumptions:
– Probability of obtaining girl .5 and boy .5
– Sexes of successive children independent
Assign Digits (or flip coin):
– 0, 1, 2, 3, 4 = girl
– 5, 6, 7, 8, 9 = boy
Simulate using line 130 of table B:
Conclusions:
Homework
 6.1-4,
8-9, 12