Chapter 17 – Probability Models
Multiple Choice Test
 Suppose we have a multiple choice test where each
question has 5 choices.
 If we had a 4 question test, find the probability that:

We get all 4 wrong

The first one we get correct is the 4th
Multiple Choice Test continued
 Suppose we have a multiple choice test with 4 questions,
each with 5 choices.
Find the probability we get:
 Only the first one correct
 Only the second correct
 Only the third correct
 Only the fourth correct
 Exactly one of the 4 correct
Bernoulli Trials
 2 possible outcomes (success and failure)
 Probability of success, p, is constant
 Trials are independent
 Often we don’t technically have independence, but if the
sample is less than 10% of population, it’s ok to use Bernoulli
trials
Binomial Model
 Binomial model used to count the number of
successes in a fixed number of Bernoulli trials
 Binom(n, p) determined by number of trials, n, and
probability of success, p.
# of ways to have k successes
 If we have 1 success in n trials, it could be the first,
second, … all the way up to the nth.
 If we have 2 successes, think about all the ways this
could happen: first 2, second 2, first and third,…
 # of ways to have k successes in n trials:
n!
n Ck 
k ! n  k !
n!  n  (n  1)  ...  3  2 1
Combinations
n!
n Ck 
k ! n  k !
 5C2

10C3
 8C6
 8C2
n!  n  (n  1)  ...  3  2 1
Binomial Model
Binom(n,p) :
 P(X = x) = nCx px qn-x
p = P(success)
q = P(failure) = 1 - p
x successes in n trials
 Expected Value: µ = np
 Standard Deviation: σ = npq
Multiple Choice Exam again
 10 questions, each question has 5 choices
 Binom(10, 0.2)
 Bernoulli trial?
 2 outcomes?
 P(success) same for all trials?
 Independent trials?/10% condition?
 P(X = 7)
P(X ≥ 7)
Multiple Choice exam cont’d.
 Binom(10, 0.2)
 Expected number of questions correct:
 Standard deviation:
Using the Normal Model to Estimate the
Binomial Model

Binom(5, 0.2)
Figures from Intro Stats, De Veaux
Binom(50, 0.2)
More with the Normal Model
 Binom(50, 0.2) centered and magnified looks like Normal(10, 2.8)
 For a large enough number of trials, the Normal model is a close
enough approximation
 Success/Failure condition: Binomial model can be
approximated by Normal if we expect at least 10 successes and
10 failures
Figure from Intro Stats, De Veaux
Using the Normal Model for a Binomial Model
 Suppose an Olympic archer can hit the bull’s-eye
80% of the time. Assume each shot is independent
of the others.
 If the archer shoots 200 arrows in a competition,
 What are the mean and standard deviation of the number of
bull’s-eyes?

What is the probability that the archer hits more than 140
bull’s-eyes?
Example from HW section, Intro Stats, De Veaux
Colorblindness Example
 About 8% of males are colorblind. A researcher
needs some colorblind subjects for an experiment
and begins checking potential subjects.
 On average, how many men should the researcher
expect to check to find one who is colorblind?
 What’s the probability that the researcher won’t find
anyone colorblind in the first 4?
Example from HW section, Intro Stats, De Veaux
Genetic trait in frogs example
 A specific genetic trait is usually found in 1 of every 8
frogs. He collects a dozen frogs. What’s the
probability that he finds the trait in:
 None of the 12 frogs?
 At least 2 frogs?
 3 or 4 frogs?
Example from HW section, Intro Stats, De Veaux