Chapter 17
Probability Models
By: Alex Woerner, and Nils Kosmo
Bernoulli Trials
● The basis for all probability models in this chapter
● Bernoulli trials have:
1. Two possible outcomes (success and failure)
2. p, the probability of success as a constant
3. Independent trials
Geometric Probability Model
● Tells us the probability for a random variable that counts
the number of Bernoulli trials until the first success.
● Completely specified by one parameter, p
● Denoted Geom(p)
● p = probability of success
● q = 1 - p = probability of failure
● X = # of trials until the first success occurs
P(X = x) = qx-1p
Independence
● Bernoulli trials must be independent.
● Bernoulli trials are not independent when we don’t
have an infinite population.
● 10% Condition: Bernoulli trials must be independent. If
this assumption is violated, it is ok to proceed as long
as the sample is smaller than 10% of the population.
Binomial Model
● Tells the probability for a random variable that counts
the number of successes in a fixed number of trials.
● Two parameters: n, number of trials, and, p,
probability of success. Denoted Binom(n,p)
n
Ck = n!/k!(n-k)!
Binomial Model Cont.
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●
●
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n = number of trials
p = probability of success
q = 1-p = probability of failure
X = # of successes in n trials
P(X = x) = nCx px qn-x
Normal Model
● Used when dealing with large numbers of trials
● Success/Failure Condition must hold to use Normal
Model
● Success/Failure Condition: Expect at least 10
successes and 10 failures
np>10 and nq>10
Continuous Random Variable
● When using Normal Model to approx. Binomial Model,
we are using a continuous random variable to approx.
a discrete random variable.
● When using Normal Model, we no longer calculate the
probability that the random variable equals a
particular value, but only that it lies between two
values.
Homework Problem #13
Homework Problem #15
Work/Answers for #13 & #15