Probability Rules!!!!!!

Probability Probability Rules!!!!!!Rules!!!!!!

ProbabilityProbabilityDefinition: a measure of how

likely it is that some event will occur. It can be expressed as a ratio of the number of favorable outcomes (i.e. a head coming up) over the total number of cases.

Spinning Pennies!Spinning Pennies!Place the penny on its edge

Lincoln side facing you.Lightly hold the penny with one

hand and flick it with the other.Let it spin until falling.Record the outcome.Run 30 trials.We’ll collect class data.

Suppose you’d only spun the penny twice….. Would our results look different?

Why do you know the proportion of heads should be 50% of the number of trials?

You will win a prize if you toss a coin a given number of times and get between 40% and 60% heads. Would you rather toss the coin 10 times or 100 times?

You will win a prize if you toss a coin a given number of times and get exactly 50% heads. Would you rather toss the coin 10 times or 100 times?

Trials, Outcomes & EventsTrials, Outcomes & Events

Trial◦Definition: A single attempt of a

random phenomenon. Outcome

◦Definition: the outcome of a trial is the value measure, observed, or reported for an individual instance of that trial.

Event◦Definition: a collection of outcomes.

Experimental ProbabilityExperimental Probability

Definition: the ratio of the number of successes over the total number of trials.

For example…the number of heads you got in your experiment divided by the total number of coin flips.

It is experimental because it comes from the actual results of your experiment.

Experimental ProbabilityExperimental Probability

Experimental probability can be expressed as:

where f is the frequency (# of successes) and n is the number of trials.

Who can provide an example of experimental probability?

n

fP

RandomnessRandomnessA phenomenon is random if we

know what outcomes could happen, but not which particular values did or will happen.

Probability is the art of summarizing this randomness using theoretical probability.

Sample SpaceSample SpaceDefinition: the sample space is the

collection of all possible outcomes from a given event.◦ Example: If the event is tossing a coin

once, then the sample space is

S = { H, T }

◦ Example: If the event is rolling a die once, then the sample space is

S = { 1, 2, 3, 4, 5, 6 }

Theoretical ProbabilityTheoretical ProbabilityDefinition: The actual probability of

an event occurring. ◦ Only works when each outcome in a

sample space has an constant defined chance of occurring.

Can anyone think of an example of theoretical probability?

space sample in the elements ofnumber

event desired in the elements ofnumber P

Probability RulesProbability Rules

1. A probability is a number between 0 and 1.

2. Something Has To Happen Rule. The probability of SOMETHING happening is always 1.

3. Complement Rule. The probability of an event occurring is 1 minus the probability that it doesn’t occur.

Disjoint EventsDisjoint EventsTwo events are disjoint if they

share no outcomes in common. If A and B are disjoint, then

knowing that A occurs tells us that B cannot occur.

Disjoint events are also called “mutually exclusive”.

The blood type, think Venn diagram where circles can’t overlap

Independent EventsIndependent EventsDefinition: if the outcome of one

event does not affect the probability of a second event the two events are considered independent.◦Ex: We are going to perform the

following experiment. First flip a coin, second roll a die. Regardless of what outcome I get for the coin flip, I will still have a 1/6 chance of rolling any one particular side of the dice. not disjoint, b/c can happen at same time

Probability RulesProbability Rules4. Addition Rule. For two disjoint

events A and B, the probability that one or the other occurs is the sum of the probabilities of the two events.

P(A or B) = P(A) + P(B) ,provided that A and B are disjoint.

5. Multiplication Rule. For two independent events A and B, the probability that both A and B occur is the product of the two events.

P(A and B) = P(A)*P(B)

What can go Wrong?What can go Wrong?Don’t confuse “disjoint” with

“independent”.◦Disjoint events cannot happen at the

same time. When one happens, you know that other did not.

◦Independent events must be able to happen at the same time. When one happens, you know it has no effect on the other.

Probability practiceProbability practiceHere are the probabilities that

the person you choose will have blood type O, A, or B. O = .49, A = .27, B = .20

What is the probability of someone with the remaining blood type AB?

What is the probability that the person chosen has either type A or type B blood?

A generic M&M bag had the following colors and probabilities of getting that color: brown=.3; red=.2, yellow=.2; green=.1; Orange=.1; What does P(blue)=?

What is the probability that the candy drawn is not red, yellow, or orange?

Create the sample space for a couple having 3 children, assume 50/50 boy/girl.

Find the probability that when a couple has 3 children, they will have exactly 2 boys. Assume that boys and girls are equally likely and that gender for any child is not influenced by the gender of any other child!

In a large Intro to Stats class, the professor reports that 55% of the students have never taken Calculus, 32% have taken only one semester, and the rest have taken two or more. The prof randomly assigns students to groups of three to work on a project. What is the probability that the first group mate you meet has studied…

Two or more semesters of Calc?Some Calc?No more than one semester of

Calc?

Another project: You are assigned to be part of a group of three students from the Intro to Stats class, from the last problem. What is the probability that of your other two group mates,

Neither studied Calc?Both have studied at least one

semester of Calc?At least one has some Calc?

Final Project: You used the Multiplication Rule to calculate the probabilities about the Calculus background of your stats group mates from the last problem.

What must be true about the groups in order to make that approach valid?

Do you think this assumption is reasonable? Why?

Opinion polling interviewers are able to reach about 65% of US households, of those contacted those that participate has fallen from 48% in 1997 to only 36% in 2003. Each house is independent of the other…

What is the prob. (2003) of the next house on the list will be contacted, but refuse to cooperate?

What is the prob. (2003) of failing to contact a house or of contacting the house but not getting them to agree to the interview?

In 1997 the contact rate was 48% and in 2003 it was 36%.

However the cooperation rate was just used: 58% in 1997 and 38% in 2003.

What is the prob. (2003) of obtaining an interview with the next call? (contacting house and agree)

Was it more likely to obtain an interview from a randomly selected house in 1997 or 2003?

The Red Cross says that about 35% of the US population has Type O blood, 30% Type A, 18% Type B, and the rest AB

Someone volunteers to give blood. What is the probability that this donor◦Has type AB blood?◦Has type A or Type B?◦Is not type O

The Red Cross says that about 35% of the US population has Type O blood, 30% Type A, 18% Type B, and the rest AB

Among 4 potential donors, what is the prob that..◦All are type O?◦No one is type AB?◦They are not all Type A?◦At least one person is Type B?

A slot machine has three wheels that spin independently. Each has 10 equally likely symbols: 3 bars, 4 lemons, 2 cherries, and a bell. If you play what is the probability

You get three lemons?You get a bell or a cherry on each?You get no fruit symbols?You get 3 bells (jackpot)?You get no bells?You get at least one bar (lose

automatically)?

For a sales promotion, the manufacturers of Coke places winning symbols under the caps of 8% of all Coke bottles. You buy a six-pack. What is the probability that you win at least once?

You shuffle a deck of cards, and then start turning them over one at a time. The first one is read. So is the second. And the third. In fact, you are surprised to get 10 red cards in a row!

You start thinking, “The next cards is due to be black!”

Are you correct in thinking that there’s a higher probability that the next cards will be black than red? Let’s discuss

Dependent EventsDependent EventsDefinition: when the outcome of one

event does affect the possible outcomes of another event, the events are called dependent events.◦ Ex: You have 5 cards in a jar. Two of them

have a picture of a Giraffe on them. The other three have a picture of a Blue Whale on them. What is the probability that the first two cards I select are both cards with Giraffes on them one after the other?

◦ LOOK AT THE SAMPLE SPACE! Make a table or chart!

Sample Spaces… Sample Spaces… For each of the following, list

the sample space and tell whether you think the outcomes are equally likely.

Toss 2 coins; record the order of heads and tails

A family has 3 children; Record the number of boys

A family has 3 children; record the genders in order of birth

Roll two dice; record the greater number


6. General Addition Rule. For any two events, then the probability of A or B is…

P(A U B) = P(A) + P(B) – P(A ∩ B)

7. Conditional Probability. The probability of B given that A happens is…

P(B|A) = )(

)(

AP

BAP

General Addition RuleReal estate ads suggest that

64% of homes for sale have garages, 21% have swimming pools, and 17% have both features. What is the probability that a home for sale has

A pool or a garage?Neither a pool nor a garage?A pool but no garage?

General addition RuleGeneral addition RuleIf we searched through the book to

check out the number of pages with a data display on it we’d find: 48% some kind of display, 27% had an equation, 7% had both data display and equation

Display these results in a Venn diagram

Find prob: random page had neither a display or equation.

Find prob: random page had a data display but no equation.

Police report that 78% of drivers stopped on suspicion of drunk driving are given a breath test, 36% a blood test, and 22% both tests. What is the probability that a randomly selected DWI suspect is given

Create a Venn diagram to display these results.

A test?A blood test or breath test, but

not both?Neither test?

In How Americans Exercise, a study presented the following information on what people do to stay fit: 53% jog, 44% swim, 46% cycle, 18% jog and swim, 15%swim and cycle, 17% jog and cycle, and 7% jog, swim, and cycle.

Find prob that a person jogs or swims.Cycles or jogsDoes neither swimming nor cycling.Does only one of the three exercises.Are there any people in the study who

don’t exercise?


6. General Addition Rule. For any two events, then the probability of A or B is…

P(A U B) = P(A) + P(B) – P(A ∩ B)

7. Conditional Probability. The probability of B given that A happens is…

P(B|A) = )(

)(

AP

BAP

Conditional ProbabilityConditional Probability

P(sports)P(Sports|Girl)P(Girl|Popular)P(Sports|Boy)P(Boy|Grades)

Goals

Grades Popular Sports Total

Gender Boy 117 50 60 227

Girl 130 91 30 251

Total 247 141 90 478

Conditional ProbabilityConditional ProbabilityFirst Second Third Crew Total

Alive 203 118 178 212 711

Survival Dead 122 167 528 673 1490

Total 325 285 806 885 2201

•P(Second Class|Dead)

•P(Dead|Second Class)

•P(Crew|Alive)

•P(Alive|First)

Survey assessing who’s been Survey assessing who’s been to what Boston landmarkto what Boston landmark

Museum of Art

Aquarium

Fenway

total

Female

20 32 28 80

Male 18 24 28 70

Total 38 56 56 150•What proportion of students who have been to the Aquarium are female?•What proportion of students who have been to Fenway are female?•What proportion of students who have been to the Museum of Art are female?•Do you think one landmark has a bigger draw than others for the female population? Or can we make that conclusion?•What is the probability that a male has gone to the Aquarium?

People who have seen James Cameron’s People who have seen James Cameron’s movies; random studymovies; random study

freshman sophomore junior senior total

Solaris 24 30 32 36 122

Point Break 12 15 20 26 73

Strange Days 10 15 18 18 61

Total 46 60 70 80 256

•What is the probability that someone has seen Point Break?•What is the probability a freshman has seen Point Break?•What’s the probability of those people who have seen Point Break we select a freshman?•What would be the probability that a senior has seen Point Break?•Do you think number of movies watched depends on age?•Are there any lurking variables?

Survey asking about Pizza Survey asking about Pizza PreferencePreference

meat vegetable cheese total

female 180 200 150 530

male 210 140 215 565

total 390 340 365 1095

•Find the probability that cheese was selected given the customer was a female.•Find the probability that cheese was selected given the customer was a male.•What is the probability that cheese pizza was selected in this study.•What question would give us the highest probability?•How much higher is the proportion of male’s that order meat than that of female’s?

Conditional ProbabilityConditional ProbabilityIn Brighton the probability that a

married man drives is .90. If the probability that a married man and his wife both drive is .85, what is the probability that his wife drives given that he drives?

Conditional ProbabilityConditional ProbabilityJackie often speeds while driving

to school in order to arrive on time. The probability that she will speed to school is .75. If the probability that she speeds and gets stopped by a police officer is .25, find the probability that she is stopped, given that she is speeding.

Conditional ProbabilityConditional ProbabilityJohn likes to study. The

probability that he studies and passes his math test is .80. If the probability that he studies is .83, what is the probability that he passes the math test, given that he has studied?


8. General Multiplication Rule. For any two events A and B (probably

dependent), the probability of A and B is

P(A∩B) = P(A)*P(B|A)

Independence AgainIndependence AgainEvents A and B are independent

if and only if…

P(B|A) = P(B)

What can go Wrong?What can go Wrong?Don’t use a simple probability

rule where a general rule is appropriate. ◦Don’t assume independence without

reason to believe it. Don’t assume outcomes are disjoint without checking.

◦Remember…the general rules always apply even when outcomes are in fact independent or disjoint.

Multiplication RuleMultiplication RuleIn the land of OZ the probability

that a man over 40 is overweight is .42. The probability that his blood pressure is high given that he is overweight is .67. If a man over 40 years of age is selected at random, what is the probability that he is overweight and that he has high blood pressure?

Multiplication RuleMultiplication RuleDue to the rising cost of auto

insurance, the probability that a randomly selected driver in Boston drives uninsured is 0.13. Moreover, the probability that the car’s driver is under 30 years old, given that the car is uninsured, is 0.39. If a driver is randomly selected, find the probability that the driver is under 30 and the car is uninsured.

Multiplication RuleMultiplication RuleA new cleaning product 410 has

recently been introduced and is being advertised on TV as having explosive cleaning qualities. The manufacturer believes that the probability people will see the commercial between noon and 4PM is 4/11. Furthermore, IF the people see the commercial, it’s SO good that the probability after seeing it they buy the product 410 is 22/36. What is the probability that the people selected at random will watch TV and buy the product?

Multiplication & IndependenceMultiplication & Independence

Two cards are drawn from a deck of 52 cards. Find the probability that both cards drawn are aces if the first card◦Is not replaced before the second

card is drawn◦Is replaced before the second card is

drawn.

The probability that a senior at BHS will go on to a college/university right away from graduation is 87%. The chance of the senior graduating college in 4 years, given that they went to a college/university right away is 49%. Find the probability that a senior from BHS goes to a college/university right from graduation and graduates in 4 years.

A bag contains: 5 red marbles, 9 blue A bag contains: 5 red marbles, 9 blue marbles, and 3 white marbles.marbles, and 3 white marbles.

If you pull three marbles without replacing them, what is the probability that:

All three are red?First two are blue, and the third

is white?Red, white, then blue?At least one red?Exactly two blue marbles?

Celtics are giving away t-shirts: Celtics are giving away t-shirts: Lucky has 5 mediums, 6 XL, and 3 Lucky has 5 mediums, 6 XL, and 3 smallssmallsHe tosses them into the crowd,

what’s the probability that:His first two tossed are both

mediums?In his first three tosses no XLs

are tossed?He tosses a small, then a

medium, then an XL?

Independence and/or Disjoint is Independence and/or Disjoint is jumpin?jumpin?Police report that 78% of drivers

stopped on suspicion of drunk driving are given a breath test, 36% a blood test, and 22% both tests.

Are giving a DUI suspect a blood test and a breath test mutually exclusive?

Are giving the two tests independent?

HMMMMmmmm….HMMMMmmmm….The publisher has the actual results of

what’s on how many pages: 48% some kind of display, 27% had an equation, 7% had both data display and equation

Make a contingency tableAre having an equation and having a

data display disjoint?Are having an equation and having a

display independent?

Tree Diagram, is it SpringTree Diagram, is it Spring

Helps organize possible outcomes for the multiplication rule.

Shows sequences of event.Illustrates conditional probability

with each branch.

Tree DiagramTree DiagramMark is flying from Boston to Denver

with a connection in Chicago. The probability his first flight leaves on time is .25. If the flight is on time, the probability that his luggage will make the connection flight in Chicago is .9, but if the first flight is delayed, the probability that the luggage will make it is only .65.

Are the first flight leaving on time and the luggage making the connection independent events? Why?

What is the probability that his luggage arrives in Denver with him?

What is the probability that his first flight was delayed given that his luggage arrived?

Tree DiagramsTree DiagramsDr. Carey has two bottles of sample pills on

his desk for the treatment of arthritic pain. One day he gives Matt a few pills from one of the bottles. (all other treatments have failed) However he does not remember from which bottle he took the pills. The pills in Bottle B1 are effective 60% of the time. The pills in bottle B2 are effective 82% of the time. B1 is closer to Dr. Carey on his desk and the probability is 3/5 that he selected the pills from this bottle, and 2/5 he selected the pills from B2 because they’re farther away on his desk.

Find the probability the pills workedWhat is the probability the pills came from

B1 given that they worked?

Tree diagramsTree diagramsAAA discovered that 44% of drivers

under the age of 30 admit to texting while driving, 37% having done it but try not to, and 19% will not while driving. An insurance company did a study that among texting drivers 17% have been involved in a texting related accident, while among those who try not to, only 9% have been involved in such accidents. Still those who don’t text get distracted by loud music and bump bumpers 2.5%

What is the probability that a randomly selected driver will be in an accident?

What is P( always texting I accident)?

reviewreview

Table shows at one point American’s view:

P(favors death penalty)P(Democrat given favor of dp)P(Republican or favor of death

penalty)

Death Penalty

Favor Opposed

Party Democrat .26 .04

Republican .12 .24

other .24 .10

reviewreview70% of kids who visit a doctor

have a fever, and 30% of kids with fever have sore throats. What’s the probability that a kid who goes to the doctor has a fever and a sore throat?

Cards: Picking three cards without Cards: Picking three cards without replacementreplacementYou get no aces?

You get all hearts?

The third card is your first red card

You have at least one diamond.

reviewreview56% of all American workers have a

workplace retirement plan, 68% have health insurance, and 49% have both benefits. We selected a worker at random…

P( a worker has neither health insurance nor a retirement plan)

P( has health insurance if he has retirement)

Are these two events independent? Mutually exclusive?

Test ReviewTest ReviewSuppose that 23% of adults smoke

cigarettes. It’s known that 79% of smokers and 13% of nonsmokers develop a certain lung condition by age 60.

What’s the probability that someone develops a lung condition by age 60?

What’s the probability that someone with a lung condition was a smoker?

Test ReviewTest ReviewOne year for Teacher Torture after the

first day Ms Forger had 12 quarters, 20 nickels, and 13 dimes.

Find the probability that the first three coins counted were all nickels

Find the probability that of the first 4 coins counted the first dime was the fourth coin.

How likely would it be to get at least one quarter in the first four coins counted?

Test ReviewTest Review

What’s the probability that a man has both conditions?

What’s the probability that he has high blood pressure?

What’s the probability that a man with high BP has high cholesterol?

Are these events independent?

Blood Press

OK high

Cholest High .11 .21

OK .16 .52

Test ReviewTest ReviewBHS guidance tells us that 72% of

seniors have had a math teacher retire from BHS, 48% have had Mrs. Belmosto, and 28% have had both. What’s the probability that a senior has

Had a teacher retire, but not Mrs. Belmosto?

A teacher retire or Mrs. Belmosto, but not both?

Neither a teacher retire nor Mrs. Belmosto?

Test reviewTest reviewYou roll a fair die three times.

What is the probability thatYou roll all 6’sYou roll al odd numbers?None of your rolls gets a number

divisible by 7?The numbers you roll are not all

5’s?

Documents

Probability Rules!!!!!!