Discrete Probability Model Calculators
Use the calculators below to compare and contrast the binomial, Poisson, geometric,
and negative binomial probability distribution models. These calculators illustrate which of these
probability models should be chosen for a given probability problem.

Binomial Distribution: used to estimate the number of successes in
a fixed number of trials
If you have
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
70
80
90
100
trials of a random process, and each trial can be classified as having one of
two outcomes (success or failure), and the probability for success is
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100
% and is the same for each trial, and the trials are independent,
then the probability of getting exactly
0
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
70
80
90
100
successes in
10
trials
is
24.609 %. The average number of successes will be
5
with an expected deviation of plus or minus
1.581 .
Poisson Distribution: used to estimate how often an event occurs
within a specified time or space
If an event occurs on average
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
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80
90
100
times per
centimeter
century
day
foot
gallon
hour
inch
kilometer
liter
meter
mile
millimeter
minute
month
second
week
yard
year
, and the events occur independently of each other, and no two
events can happen at exactly the same time, then the probability that the event
will happen exactly
0
1
2
3
4
5
6
7
8
9
10
11
12
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30
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50
60
70
80
90
100
times in an
hour
is
3.783 %. On average
this event will occur
10.000
times in an
hour
with an expected deviation of plus or minus
3.162 .
Geometric Distribution: used to estimate the number of trials that
must occur before the first success
If you are conducting trials of a random process, and each trial can be classified
as having one of two outcomes (success or failure), and the probability for success
is
1
2
3
4
5
6
7
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10
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100
% and is the same for each trial, and the trials are independent,
then the probability of getting the first success on trial number
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
70
80
90
100
is
25.000 %. On average,
a single success will occur in
2.000
trials with an expected deviation of plus or minus
1.414 .
Negative Binomial Distribution: used to estimate the number of
trials that must occur before the kth success is observed
If you are conducting trials of a random process,
and each trial can be classified as having one of two outcomes (success or failure),
and the probability for success is
1
2
3
4
5
6
7
8
9
10
11
12
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14
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99
100
% and is the same for each trial, and the trials are independent,
then the probability of getting the
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2
3
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5
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7
8
9
10
11
12
13
14
15
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20
30
40
50
60
70
80
90
100
th success on the
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
70
80
90
100
th trial is
8.810 %. On average, there will be
20.000
trials before the
10 th
success is observed with an expected deviation of plus or minus
4.472 .