Discrete probability Discrete probability distributionsdistributions

3)For every possible x value,

0 < P(x) < 1.

4) For all values of x,

S P(x) = 1.

Think About It…

1. In a game of dice a friend gives you a choice. If an even number is rolled you win $100, if a 5 is rolled you win $100. Which option would you choose? Why?

1. You friend now states if a 5 is rolled you will win $200. Which option would you choose? Why?

2. Again, your friend increased a roll of 5 to $300.Which option would you choose? Why

1. Finally your friend states a roll of 5 will win you $400. Which option would you choose? Why?

Random Variable -Random Variable -

A numerical variable whose value depends on the outcome of a chance experiment

Two types:Two types:

Discrete – count of some random variable

Continuous – measure of some random variable

Random Variable

Example:

Consider tossing a fair coin 3 times.Define X = the number of heads obtained

The probabilities pi must satisfy two requirements:

1.Every probability pi is a number between 0 and 1.

• 0 < P(x) < 1.

•The sum of the probabilities is 1.

1. Σ P(x) = 1.

To find the probability of any event, add the probabilities pi of the particular values xi that make up the event.

Discrete

Discrete Probability Discrete Probability DistributionDistribution

A distribution of a random variable gives its possible values and their probabilities.

1)Usually displayed in a table, but can be displayed with a histogram or formula

Probability Distribution

Example:

Consider tossing a fair coin 3 times.Define X = the number of heads obtained

Make a Probability Distribution Chart

Make a Probability Distribution Histogram

What are the chances

What is P(X > 2)

Show that this is a legitimate probability distribution.

Let x be the number of courses for which a randomly selected student at a certain university is registered.

X 1 2 3 4 5 6 7 P(X) .02 .03 .09 ? .40 .16 .05

P(x = 4) =

P(x < 4) =

P(x < 4) =

What is the probability that the student is registered for at least five courses?

Why does this not start at zero?

.25

.14

.39 P(x > 5) = .61

Example: Babies’ Health at Birth

Read the example on page 343.

(a) Show that the probability distribution for X is legitimate.

(b) Make a histogram of the probability distribution. Describe what you see.

(c) Apgar scores of 7 or higher indicate a healthy baby. What is P(X ≥ 7)?

(a) All probabilities are between 0 and 1 and they add up to 1. This is a legitimate probability distribution.

(b) The left-skewed shape of the distribution suggests a randomly selected newborn will have an Apgar score at the high end of the scale. There is a small chance of getting a baby with a score of 5 or lower.

(c) P(X ≥ 7) = .908We’d have a 91 % chance of randomly choosing a healthy baby.

Value: 0 1 2 3 4 5 6 7 8 9 10

Probability:

0.001

0.006

0.007

0.008

0.012

0.020

0.038

0.099

0.319

0.437

0.053

Formulas for mean & Formulas for mean & variancevariance

ixix

iix

px

px

22

Found on formula card!

Dice

Expected Value Comparisons

Tebow Time!The NFL Draft is an annual event which is the most

common source of player recruitment. In the first round of the 2010 NFL draft the Denver Broncos selected Tim Tebow. At the position of Quarterback Tebow’s ability was highly debated on a national level. The Broncos’ Franchise took a major risk, however, do no think for a second this was not a calculated risk.

Imagine you are on the Broncos Management.Judging by his record in College, analysts

predict Tebow has a 10% chance of becoming an elite quarterback, pulling in $20 million for the franchise. He has a 40% chance of being average, bringing in $10 million. Otherwise, he will be 2nd or 3rd string which brings in no money and would be a loss (the cost of the contract) of $9.7 million.

Example: Apgar Scores – What’s Typical?

Consider the random variable X = Apgar Score

Compute the mean of the random variable X and interpret it in context.

Value: 0 1 2 3 4 5 6 7 8 9 10

Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053

The mean Apgar score of a randomly selected newborn is 8.128. This is the long-term average Agar score of many, many randomly chosen babies.

Note: The expected value does not need to be a possible value of X or an integer! It is a long-term average over many repetitions.

You Try

…

1. A college instructor teaching a large class traditionally gives 10% A’s, 20% B’s, 45% C’s, 15% D’s, and 10% F’s. If a student is chosen at random from the class, the student’s grade on a 4-point scale (A = 4) is a random variable X. Create the probability distribution of X.

What is the probability that a student has a grade point of 3 or better in this class?

Draw a probability histogram to picture the probability distribution of the random variable X.

2. Put all the letters of the alphabet in a hat. If you choose a consonant, I pay you $1. If you choose a vowel, I pay you $5. X is the random variable representing the outcome of the experiment.

Create the distribution of X

What is your expected payoff (value) in this game?

2. Put all the letters of the alphabet in a hat. If you choose a consonant, I pay you $1. If you choose a vowel, I pay you $5. X is the random variable representing the outcome of the experiment.

What is the game’s variance? The Standard deviation?

Let x be the number of courses for which a randomly selected student at a certain university is registered.

X 1 2 3 4 5 6 7

P(X) .02 .03 .09 .25 .40 .16 .05

What is the expected value and standard deviations of this distribution?m = 4.66 & s = 1.2018

Is the formula the only way?!?!?!?!!?

Stat, 1:EditL1 = Random Variable ( X )

L2 = Probability (pi)

Stat, Calc, 1: 1-Var Stats2nd Stat L1 2nd Stat L2

1-Var Stats L1, L2

Let x be the number of courses for which a randomly selected student at a certain university is registered.

X 1 2 3 4 5 6 7

P(X) .02 .03 .09 .25 .40 .16 .05

What is the expected value and standard deviations of this distribution?m = 4.66 & s = 1.2018

.

Box of 20 DVDs, 4 are defective. Select two from the box without replacement

•Identify your random variables.

•Create a Probability Distribution

•What is the mean (expected value) of the discrete

random variable?

•What is the variance? The Standard Deviation?

Cars in a TownX = number of vehicles owned by a household in a random town

P(0) =.05, P(1) = .45, P(2) = .275, P(3) = .1, P(4) = .075, P(5) = .05

•Identify your random variables.

•Create a Probability Distribution

•What is the mean (expected value) of the discrete

random variable?

•What is the variance? The Standard Deviation?

Book Editor X = # of errors that appear on a randomly selected page of a bookX = 0, 1, 2, 3 ,4

P(0) =.73, P(1) = .16, P(2) = .06, P(3) = .04, P(4) = .01

•Identify your random variables.

•Create a Probability Distribution

•What is the mean (expected value) of the discrete

random variable?

•What is the variance? The Standard Deviation?

Flights from LA to ChicagoX = # of flights that are on time out of 3 independent flights

P(0) =.064, P(1) = .288, P(2) = .432, P(3) = .216

•Identify your random variables.

•Create a Probability Distribution

•What is the mean (expected value) of the discrete

random variable?

•What is the variance? The Standard Deviation?

Linear combinationsLinear combinations

222

21

21y

21

...

...

then...If

xnxxy

xnxx

nxxxy

Just add or subtract the means!

If independent, always addadd the variances!

A nationwide standardized exam consists of a multiple choice section and a free response section. For each section, the mean and standard deviation are reported to be

mean SD

MC 38 6

FR 30 7

If the test score is computed by adding the multiple choice and free response, then what is the mean and standard deviation of the test?

m = 68 & s = 9.2195

Linear function of a Linear function of a random variablerandom variable

If x is a random variable and a and b are numerical constants, then the random variable y is defined by

and

bxay

xyxbxay

xbxay

bb

ba

or2222

The mean is changed by addition &

multiplication! The standard deviation is ONLYONLY

changed by multiplication!

Let x be the number of gallons required to fill a propane tank. Suppose that the mean and standard deviation is 318 gal. and 42 gal., respectively. The company is considering the pricing model of a service charge of $50 plus $1.80 per gallon. Let y be the random variable of the amount billed. What is the mean and standard deviation for the amount billed?m = $622.40 & s = $75.60