4.5 – Comparing Probability Distributions
In earlier chapters, you looked at independent and dependent events. When determining the
appropriate probability distribution, this is an important criterion to consider. How would you
differentiate between independent and dependent events?
The following chart summarizes the general conditions of the distributions:
Parameters and
what they represent
Definition of random
variable, x
Range of values for x
Probability formula
Expectation formula
Identifying
characteristics
Uniform
𝑛 = number of items
Value of the outcome
Depends on the situation
1
𝑃(𝑥) =
𝑛
𝐸(𝑋) =
1
∑𝑥
𝑛
All items are equally likely
A single trial
Binomial
𝑛 = number of trials
𝑝 = probability of success on
an individual trial
𝑞 = probability of failure on an
individual trial
Number of successful
Outcomes
𝑥 = 0, 1, 2, … , 𝑛
𝑛
𝑃(𝑥) = ( ) (𝑝)𝑘 (1 − 𝑝)𝑛−𝑘
𝑘
Hypergeometric
𝑛 = size of the population
𝑟 = number of trials
𝑎 = number of successful
items available
𝐸(𝑋) = 𝑛𝑝
𝐸(𝑥) =
Trials are independent
Successes are counted
Number of successful
Outcomes
𝑥 = 0, 1, 2, … , 𝑟
(𝑎)(𝑛−𝑎)
𝑃(𝑥) = 𝑥 𝑛𝑟−𝑥
(𝑟 )
𝑟𝑎
𝑛
Trials are dependent
Successes are counted
Compare and contrast the general conditions for the binomial and hypergeometric probability
distributions:
Compare and Contrast
Concept 1
Binomial Distribution
Concept 2
Hypergeometric Distribution
How are they alike?
Random variable, x, is the number of successes.
Fixed number of trials.
Successes are counted.
Distributions is discrete
How are they different?
Number of trials
Independent trials
Size of population
Dependent trials
𝑥 = 0, 1, 2, … , 𝑛
Probability of success does not change
𝐸(𝑋) = 𝑛𝑝
𝑥 = 0, 1, 2, … , 𝑟
Probability of success changes in every trial
𝑟𝑎
𝐸(𝑥) =
𝑛
Parameters: 𝑛, 𝑝, 𝑞
Parameters: 𝑛, 𝑟, 𝑎
Mathematics of Data Management (MDM4UC)
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4.5 – Comparing Probability Distributions
Use the following flow chart to help you decide when to use each type of distribution (uniform,
binomial, hypergeometric, neither).
Practice: (Page 185) #1-4, 6
Mathematics of Data Management (MDM4UC)
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