Probability Models

Chapter 17 AP Stats

Bernoulli Trials

In order for a situation to qualify for being considered a Bernoulli Trial, it must satisfy the below:

1. There are only two possible outcomes, success and failure, on each trial.

2. The probability of success is the same on every trial

3. The trials are independent

Probability Models

If a situation can be considered a Bernoulli Trial, then we can use one of the two models below—depending upon what we are trying to find and what we are given.

1. Geometric Model2. Binomial Model

Probability Models

What do we do with these models?• Find expected values (means)• Find standard deviations• Find probabilities of events occuring

Probability ModelsGeometric Model

p= probability of successq= probability of failure (1-p)x = number of trials until the first success.

p

XE1

2p

qXSD

pqxXP x 1

“How many until the first success?”

“What is the probability that the first success occurs . . . . ?”

Probability ModelsBinomial Model

p= probability of successq= probability of failure (1-p)x = number of trials until the first successn = number of trials

npXE

npqXSD

!!

! where,

xnx

n

k

nqp

k

nxXP knk

“How many successes in a given number of trials?”

“What is the probability that there are k successes in n trials”

.

ofn combinatio a is

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thingsnk

n

trials"in successes " eans nkmk

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ExampleThe probability that a student gets a perfect score on an Algebra test is

40%.

Find the expected number of students it would take until a student was randomly selected who received a perfect score.

(GEOMETRIC)

How many students would I expect to have a perfect test score out of a group of 10 students?

(BINOMIAL)

Remember, first make sure that this situation satisfies the conditions for being a Bernoulli Trial!!

Example

Find the probability of having to select 7 students before you get a perfect score.

Find the probability that not more than 2 students out of 8 get a perfect score.

How to Calculate on Calculator

Geometric—When finding the probability of an individual event

What is the probability that our first perfect test score comes from the 5th person we randomly select?

Use geometpdf(p,x)

How to Calculate on Calculator

Geometric—When finding the probability of several probabilities

What is the probability that our first perfect test score comes by the time we select our 5th person? (Could be 1st person, 2nd, 3rd, 4th or 5th)

Use geometcdf(p,x)---- (x: finding success on or before the xth trial).

How to Calculate on Calculator

Binomial—When finding the probability of an individual event.

What is the probability that 2 out of 8 people randomly selected have a perfect test?

Use binompdf(n,p,x)

How to Calculate on Calculator

Binomial—When finding the probability of several probabilities. (Part I)

What is the probability that less than 3 out of 8 people randomly selected have a perfect test?

Use binomcdf(n,p,x) (“getting x or fewer in n trials”)

How to Calculate on Calculator

Binomial—When finding the probability of several probabilities. (Part II)

What is the probability that more than 3 out of 8 people randomly selected have a perfect test? Think about as “not less than 4”

Use 1-binomcdf(n,p,x) (“getting x or fewer in n trials”)

Independence

Remember—For a Bernoulli Trials, the trials must be independent.

You MUST check that assumption.

Independence and the 10% Condition

What if our trials are not independent?We can pretend that they are independent and

proceed if and only if we satisfy the 10% Condition.

10% Condition: If the Independence Assumption is violated, we can still proceed as if we are dealing with a Bernoulli Trial as long as the sample is smaller than 10% of the population

Binomial Model and the Normal Curve

If we have a situation that can be modeled by a Binomial Model, we can instead, use a Normal to help us determine probabilities.

In order to do this, however, we must satisfy the Success/Failure Condition

Binomial Model and the Normal Curve

Success/Failure ConditionA Binomial Model is approximately Normal if we

expect at least 10 successes and 10 failures:10 and 10 nqnp

This condition needs to be formally checked if you use the Normal Model to approximate a Binomial Model.

Example

In the United States, the probability of having twins (usually about 1 in 90 births) rises to about 1 in 10 births for women who have been taking the fertility drug Clomid. At a large fertility clinic, 152 women became pregnant while taking Clomid. What is the probability that no more than 10 of the women have twins?