Probability Models
Chapter 17
Bernoulli Trials (Binomial Setting)
• The basis for the probability models we will examine in this
chapter is the Bernoulli trial.
• We have Bernoulli trials if:
– there are two possible outcomes (success and failure).
– the probability of success, p, is constant.
– the trials are independent.
The Binomial Model
• A Binomial model tells us the probability for a
random variable that counts the number of
successes in a fixed number of Bernoulli trials.
• Two parameters define the Binomial model: n,
the number of trials; and, p, the probability of
success. We denote this Binom(n, p).
Check
• Genetics says that children receive genes from
each of their parents independently. Each
child of a particular pair of parents has
probability 0.25 of having type O blood.
Suppose these parents have 5 children. Let
X=the number of children with type O blood.
Is this a binomial distribution?
Check
• Shuffle a deck of cards. Turn over the first 10
cards, one at a time. Let Y=the number of
aces you observe.
Is this a binomial distribution?
Check
• Shuffle a deck of cards. Turn over the top
card. Put the card back in the deck, and
shuffle again. Repeat this process until you
get an ace. Let W=the number of cards
required.
Is this a binomial distribution?
The Binomial Model
(Binomial Coefficient)
• In n trials, there are
n!
n Ck 
k ! n  k !
ways to have k successes.
– Read nCk as “n choose k.”
• Note: n! = n  (n – 1)  …  2  1, and n! is
read as “n factorial.”
The Binomial Model
Binomial probability model for Bernoulli trials:
Binom(n,p)
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
  np
  npq
The Normal Model to the Rescue!
• When dealing with a large number of trials in
a Binomial situation, making direct
calculations of the probabilities becomes
tedious (or outright impossible).
• Fortunately, the Normal model comes to the
rescue…
The Normal Model to the Rescue!
• As long as the Success/Failure Condition
holds, we can use the Normal model to
approximate Binomial probabilities.
– Success/failure condition: A Binomial model is
approximately Normal if we expect at least 10
successes and 10 failures:
np ≥ 10 and nq ≥ 10
The Geometric Model
• A single Bernoulli trial is usually not all that
interesting.
• A Geometric probability model tells us the
probability for a random variable that counts
the number of Bernoulli trials until the first
success.
• Geometric models are completely specified by
one parameter, p, the probability of success,
and are denoted Geom(p).
The Geometric Model
Four Conditions for a Geometric Setting:
1. Binary: The possible outcomes of each trial can
be classified as “success” or “failure”.
2. Independent: Trials must be independent; that
is, knowing the result of one trial must not have
any effect on the result of any other trial.
3. Trials: The goal is to count the number of trials
until the first success occurs.
4. Success: On each trial, the probability p of
success must be the same.
Situation
• Your teacher is planning to give you 10 problems for homework. As
an alternative, you can agree to play the birthday game. Here’s
how it works. A student will be selected at random from your class
and asked to guess the day of the week (for instance, Thurs) on
which one of your teacher’s friends was born. If the student
guesses correctly, then the class will have only one homework
problem. If the student guesses the wrong day of the week, your
teacher will once again select a student from the class at random.
The chosen student will try to guess the day of the week on which a
different one of your teacher’s friends was born. If this student
gets it right, the class will have two homework problems. The game
continues until a student correctly guesses the day on which one of
your teacher’s (many) friends was born.
Trial
• The random variable of interest in this game is
Y=the number of guesses it takes to correctly
match the birth day of one of your teacher’s
friends. Each guess is one trial of the chance
process. Let’s check the conditions for a
geometric setting:
The Geometric Model
Geometric probability model for Bernoulli trials:
Geom(p)
p = probability of success
q = 1 – p = probability of failure
X = number of trials until the first success occurs
x-1
P(X = x) = q p
1
E(X)   
p

q
p2
Independence
• One of the important requirements for Bernoulli trials is that
the trials be independent.
• When we don’t have an infinite population, the trials are not
independent. But, there is a rule that allows us to pretend we
have independent trials:
– The 10% condition: Bernoulli trials must be independent. If
that assumption is violated, it is still okay to proceed as
long as the sample is smaller than 10% of the population.
Continuous Random Variables
• When we use the Normal model to
approximate the Binomial model, we are using
a continuous random variable to approximate
a discrete random variable.
• So, when we use the Normal model, we no
longer calculate the probability that the
random variable equals a particular value, but
only that it lies between two values.
What Can Go Wrong?
• Be sure you have Bernoulli trials.
– You need two outcomes per trial, a constant
probability of success, and independence.
– Remember that the 10% Condition provides a
reasonable substitute for independence.
• Don’t confuse Geometric and Binomial models.
• Don’t use the Normal approximation with small
n.
– You need at least 10 successes and 10 failures to use
the Normal approximation.
What have we learned?
• Bernoulli trials show up in lots of places.
• Depending on the random variable of interest,
we might be dealing with a
– Geometric model
– Binomial model
– Normal model
What have we learned?
– Geometric model
• When we’re interested in the number of Bernoulli trials
until the next success.
– Binomial model
• When we’re interested in the number of successes in a
certain number of Bernoulli trials.
– Normal model
• To approximate a Binomial model when we expect at
least 10 successes and 10 failures.