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 correctOnly the second correctOnly the third correctOnly 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: !

! ( 1) ... 3 2 1! !

n k

nC n n n

k n k

Combinations

5C2

10C3

8C6

8C2

!

! ( 1) ... 3 2 1! !

n k

nC n n n

k n k

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 choicesBinom(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) Binom(50, 0.2)

Figures from Intro Stats, De Veaux

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