Transcript Chapter 3

Chapter 3
Section 3.2
Basic Terms of Probability
Probability
Probability is a measurement of the chance that some event is likely to happen.
We have heard many examples of this in our everyday lives.
There is a 70% chance of rain today.
The odds on that horse winning the race are 3:2.
A smoker is 10% more likely to get cancer.
The chance he makes a full house (poker hand) is 9%.
All of these things have to do with probability. In any measurement we assign a
number to something in some sort of consistent and meaningful way. In the case
of probability we assign a number to the chance an event will happen.
Consistent – Anyone following this procedure will get the same number.
Meaningful – The number is not negative and the number assigned will
increase as the chance gets larger.
In order to accomplish both of these things we use the language of sets. The
number that represents the probability will be found by counting the number of
things in two sets and dividing them.
Terminology
We will refer to certain terms that determine the sets used to calculate the
probability.
Experiment – Any situation whose outcome is uncertain. Examples would include:
roll a dice, flip a coin, pick a card, predict rain, predict disease etc.
Sample Space – All the equally likely outcomes of an experiment. (Denoted S)
Event – A subset of the sample space you are interested in.
n(A) –
This stands for the number of elements in the set A. For example if the set
C={yellow, green, blue, red} then n(C)=4.
The probability of an event E is denoted using the symbol P(E) and is given by:
n( E ) num berof elem entsin E
P( E ) 

n( S ) num berof elem entsin S
Example:
Experiment : Flip two coins
Event A: At least one is a head
S
S = {HH, HT, TH, TT}
A = {HH, HT, TH}
A
HH
HT
TH
TT
P( A) 
n( A) 3
  75%
n( S ) 4
Venn Diagrams are very useful for picturing what is going on in an experiment.
Consider the experiment a drawing one card at random from a regular deck of
cards. A regular deck has 52 cards. The cards are broken down in 13
denominations: 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack (J), Queen (Q), King (K) and Ace (A).
Each of the denominations occurs in 4 different suits: ,,,.
Find the probabilities of each of the following events.
The probability of getting an eight.
S
4
Let E be this event.
The probability of getting a red
card.
E
S
Let R be this event.
The probability of getting a red
eight.
Let RE be this event.
S
P( R) 
n( R) 26 1

  50%
n( S ) 52 2
26
RE
2
n( E ) 4
1

  7.7%
n( S ) 52 13
48
R
26
P( E ) 
50
P( RE) 
n( RE) 2
1


 3.8%
n( S ) 52 26
Outcomes that are not Equally Likely
Some experiments have outcomes that can occur repeatedly.
Example:
Experiment: Roll two dice and take the sum of both numbers rolled.
Sample Space: {2,3,4,5,6,7,8,9,10,11,12}
The problem with this sample space is that the numbers do not occur with the
same number of chances. Getting a 12 can only occur in one way with a six on
each dice, but getting a 4 can occur in three different ways. In this example we will
make a table to show all the possible outcomes.
1
2
3
4
 2.78% P(3) 
 5.56% P(4) 
 8.33% P(5) 
 11.1%
36
36
36
36
5
6
5
4
P(6) 
 13.8% P(7) 
 16.6% P(8) 
 13.8% P(9) 
 11.1%
36
36
36
36
3
2
1
P(10) 
 8.3% P(11) 
 5.56% P(12) 
 2.78%
36
36
36
P(2) 
P(Getting 7 or 11) 
8
 22.2%
36
15
P(Getting a number greater than 7) 
 41.67%
36
1 2 3 4
5
6
1 2 3 4 5
6
7
2 3 4 5 6
7
8
3 4 5 6 7
8
9
4 5 6 7 8
9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
Experiments With more than 1 Event Associated With Them
Most of the interesting questions that probability can answer have to do with more
than one event taking place. Consider the following experiment.
Experiment: Flip a coin 3 times
Sample Space: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Event F: The first coin flipped is a head.
Event A: All coins are the same.
The Venn Diagram to the right shows the
eight outcomes in the sample space and
how they fit with the two events.
4
2
P( F )   50%
P( A)   25%
8
8
1. The chance both F and A occur at the same time.
2. The chance either F or A occurs.
3. The chance F occurs and A does not.
4. The chance neither F nor A occurs.
5. The chance A does not occur.
S
F
A
HHT
HTH
HHH
HTT
1
 12 .5%
8
5
2. P( F  A)   62.5%
8
3
3. P( F  ( A' ))   37.5%
8
3
4. P(( F  A)' )   37.5%
8
6
5. P ( A' )   75 %
8
1. P ( F  A) 
TTT
THH
THT
TTH
Odds of an Event
Often in probability people talk about the odds for an event. This is related to the
probability. The odds in favor of an event E is the ratio between the number of items
in the sample space that make up the event and the number that make up the
complement. The odds in favor of E are marked with o(E) and are usually written in
ratio form with a (:).
o(E) = n(E) : n(E')
If you know the odds you can find the probability and if you know the probability you
can find the odds. Look at a previous example.
Example:
Experiment : Flip two coins
Event A: At least one is a head
What are the odds in favor of A?
S
S = {HH, HT, TH, TT}
A = {HH, HT, TH}
A
HH
HT
TH
TT
P( A) 
n( A) 3
  75%
n( S ) 4
o(A) = n(A) : n(A') = 3:1 "Read 3 to 1"
If the P(B)=⅜, what are the odds in favor of B? P(B')= 1-⅜ = ⅝, which gives o(B) = 3:5
If the o(C)=1:7, what is P(C)?
P(C ) 
n(C )
n(C )
1
1



n( S ) n(C )  n(C ' ) 1  7 8