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Probability Rules!!!!!!
Probability

Definition: 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!
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 & 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 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 Probability

Experimental probability can be expressed as:
f
P
n

where f is the frequency (# of successes) and n is
the number of trials.
Who can provide an example of experimental
probability?
Randomness
A 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 Space

Definition: 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 Probability

Definition: The actual probability of an event
occurring.
◦ Only works when each outcome in a sample
space has an constant defined chance of
occurring.
number of elements in the desired event
P
number of elements in the sample space

Can anyone think of an example of
theoretical probability?
Probability Rules
1.
2.
3.
A probability is a number between 0
and 1.
Something Has To Happen Rule.
The probability of SOMETHING
happening is always 1.
Complement Rule. The probability
of an event occurring is 1 minus the
probability that it doesn’t occur.
Disjoint Events
Two 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 Events

Definition: 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 Rules
4.
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) ,
Multiplication Rule. For two independent
events A and B, the probability that both A and
B occur is the product of the two events.
provided that A and B are disjoint.
5.
P(A and B) = P(A)*P(B)
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 practice
Here 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 Events

Definition: 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…
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

Probability Rules
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) =
P( A  B)
P( A)
General Addition Rule
 Real 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 Rule
If 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 jogs
Does neither swimming nor cycling.
Does only one of the three exercises.
Are there any people in the study who
don’t exercise?
Probability Rules
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) =
P( A  B)
P( A)
Conditional Probability
Goals
Grades
Gender Boy 117
Girl 130
Total 247





P(sports)
P(Sports|Girl)
P(Girl|Popular)
P(Sports|Boy)
P(Boy|Grades)
Popular
50
91
141
Sports
60
30
90
Total
227
251
478
Conditional Probability
First Second
Third
Crew
Total
Alive
203
118
178
212
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)
711
Survey assessing who’s been to
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 movies;
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
Preference
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 Probability

In 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 Probability

Jackie 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 Probability

John 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?
Probability Rules
8.
General Multiplication Rule. For any
two events A and B
the
probability of A and B is
(probably dependent),
P(A∩B) = P(A)*P(B|A)
Independence Again

Events A and B are independent if and
only if…
P(B|A) = P(B)
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 Rule

In 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 Rule

Due 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 Rule

A 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 & 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 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: Lucky has 5
mediums, 6 XL, and 3 smalls
He 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 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….
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 table
 Are having an equation and having a data display
disjoint?
 Are having an equation and having a display
independent?

Tree Diagram, is it Spring
Helps organize possible outcomes for the
multiplication rule.
 Shows sequences of event.
 Illustrates conditional probability with
each branch.

Tree Diagram
Mark 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 Diagrams
Dr. 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 worked
 What is the probability the pills came from B1
given that they worked?

Tree diagrams
AAA 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)?

review
Party
Death
Penalty
Favor
Opposed
Democrat
.26
.04
Republican
.12
.24
other
.24
.10
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)

review

70% 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
replacement

You get no aces?

You get all hearts?

The third card is your first red card

You have at least one diamond.
review
56% 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 Review
Suppose 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 Review
One 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 Review
Blood




Press
OK
high
Cholest High
.11
.21
OK
.16
.52
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?
Test Review
BHS 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 review
You roll a fair die three times. What is
the probability that
 You roll all 6’s
 You roll al odd numbers?
 None of your rolls gets a number divisible
by 7?
 The numbers you roll are not all 5’s?
