Sec 5.2 Part one


Sample Space:
◦ For a random Phenomenon the sample space is the set of all possible outcomes.
◦ Examples:
For the flip of two coins the sample space is.
{(HH), (HT), (TH), (TT)}
A single role of a six sided die.
{1, 2, 3, 4, 5, 6}
Types of pets people have.
{Dogs, Cats, Birds, Fish, reptiles, Insects, mollusks}


Event:
◦ An event is a subset of the sample space.
◦ Example:
For the flip of two coins an event could be two heads.
A single role of a six sided die rolling a 5 would be an event.
For pets the selection of a mammal could be an event.


You are building an experiment testing the effects of caffeine, sugar, and alcohol on memory. You are going to randomly assign each of your subjects to take one or all of these substances by flipping a coin for each.

Lets look at the sample space for our subjects.

Tree Diagram
Caffeine
Subject
None
Sugar
None
Sugar
None
Alcohol
Sample Space
CSA
None CSN
Alcohol CNA
None CNN
Alcohol
None
NSA
NSN
Alcohol
None
NNA
NNN


So with our sample space;
{CSA, CSN, CNA, CNN, NSA, NSN, NNA, NNN}
There are many different types of possible events.
Examples:
1) CNA
2) Takes caffeine
3) Does not take caffeine
4) Takes either sugar or caffeine but not alcohol
5) Takes nothing (Control)


First, some things that must be true about our probabilities.
◦ The probability of each individual outcome is between zero and one.
◦ The total of all the individual probabilities equals ONE.


The probability of an event A, denoted P(A), is the ratio between the number of outcomes in that event and the number of outcomes within the sample space.
When all possible outcomes are equally likely!!!!

The probability of an event B, denoted P(B), is found by adding the probabilities of the individual outcomes.


So back to our sample space;
{CSA, CSN, CNA, CNN, NSA, NSN, NNA, NNN}
Question:
What is the probability a subject got caffeine or sugar?
To Answer:
How Many outcomes had either caffeine or sugar?
6
How many outcomes are in the sample space?
8
So, the probability of this event is;
P ( A )

6
8

3
4
So, the probability that a given subject was given either caffeine or sugar is .75 (or 75%). Either notation is ok with me.


Again given our sample space;
{CSA, CSN, CNA, CNN, NSA, NSN, NNA, NNN}
Question:
What is the probability a subject got caffeine or sugar not both?
To Answer:
How Many outcomes had either caffeine or sugar not both?
4
How many outcomes are in the sample space?
8
So, the probability of this event is;
P ( A )

4
8

1
2
So, the probability that a given subject was given either caffeine or sugar is .5 (or 50%).


Yet again given our sample space;
{CSA, CSN, CNA, CNN, NSA, NSN, NNA, NNN}
Question:
What is the probability a subject receiving only two compounds?
To Answer:
How Many outcomes have only two compounds?
3
How many outcomes are in the sample space?
8
So, the probability of this event is;
P ( A )

3
8
So, the probability that a given subject was given either caffeine or sugar is .375 (or 37.5%).


A sample of 3000 registered voters is taken to find the probability of certain events by party affiliation. The data is summarized below.
Republicans
Democrats
Other
Total
Voted
731
514
182
1426
Not Voted
522
940
121
1572
Total
1253
1454
293
3000

Republicans
Democrats
Other
Total
Voted
731
514
182
1426
Not Voted
522
940
121
1572
Total
1253
1454
293
3000
1)
2)
3)
What is the sample space.
{(R,V), (R,NV), (D,V), (D,NV), (O,V), (O,NV)}
What is the probability that a selected subject voted?
1426/3000
What is the probability that a selected subject was a
Democrat?
1454/3000
More Questions??

5.2 Part two.


Complement of an event;
◦ The complement of an event A is made up of all outcomes in the sample space that are not within A.
A

The sum of the probabilities of an event and its complement always add to one .
 Thus, P(~A)=1- P(A)


Let A be the event of rolling a 6 on a fair die.
◦ Then ~A would be…

Let A be the event of having at least one woman on the supreme court.
◦ Then ~A would be…

The event of being in the bubble
◦ Then ~A would be...

U U
A B
The Sample Space,
“Universe”
Bubbles within are events.
“Sets”

U
A
U
~A
An event A.
The event ~A

U
A B
U
A B
Disjoint events, Non-disjoint events.

Goal: Make a Venn diagram with event A as getting one compound and event B getting two compounds.
A
CSA
CSN
CNA
NSA
B
NNA
NSN NNN
CNN
Are these disjoint events?
What is ~A?
What is ~B?

U
Given events A and B we can find the...
U
A B
A B
Intersection of A and B, Union of A and B.


The intersection of A and B is made up from events in
A
B.
◦ Is this more or less of a restriction?
◦ Is this going to make our Probabilities bigger or smaller?
◦ Link “and” with shrinking probabilities


The union of A and B is made up from events in
A
B.
◦ Is this more or less of a restriction?
◦ Is this going to make our Probabilities bigger or smaller?
◦ Link “or” with growing probabilities.

U
Given events A and B we can find the...
U
A B
A B
Intersection of A and B,
“A and B”
Union of A and B.
“A or B”


To find the probability of Event A or Event B occurring we must use unions.
Case 1: If the events are disjoint find the sum of the probabilities of events A and B.
P(A or B) = P(A) + P(B)
Case 2: If the events are not disjoint add the probabilities of the two events and subtract the probability of their intersection.
P(A or B) = P(A) + P(B) – P(A and B)

U
A
B
Take Notes on board work for complements, unions.


For independent events;

To find the probability of Event A and Event B occurring we must use intersections.
◦ The probability of being within an intersection is the product of the event probabilities.
P(A and B) = P(A) * P(B)

U
A
B
Take Notes on board work intersections and more.

Subject
Caffeine
None
Sugar
None
Sugar
None
Alcohol
None
Alcohol
None
Alcohol
None
Alcohol
None
In our experiment we made each event equally likely. What if we wanted the change this. In order to make our control group larger lets role a die instead of flipping a coin. For each compound if the die comes up four or less they get a dose. If the die is five or six they don’t get that compound.
What is the Probability of getting all three compounds?
What is the probability of getting caffeine or Alcohol?
What is the Probability of getting at least two compounds?
Let Map this out on our Hand outs.


Probabilities are between zero and one.

The sum of the all probabilities is one.

P(A) + P(~A) = 1

The union: P(A or B) = P(A) + P(B) – P(A or B)

For Independent the intersection: P(A and B) = P(A)*P(B)

Events are disjoint if they have no common element.