Statistics – Chapter 4 – Probability – Notes
I.
Probability – is the likelihood of something happening.
II.
Types of Probability
a. Theoretical (equally likely outcomes) – “What should happen”
i. P(A) = # of favorable / Total # of outcomes
b. Experimental (relative frequency) – “What actually happens”
i. P(A) = relative frequency = f / n
c. Intuition (gut feeling) – “What you feel like should happen”
III.
Sample Space
a. The sample space is a list or set of all the possible outcomes.
b. Examples:
i. The sample space for rolling a die is {1,2,3,4,5,6}, or for flipping a coin is {H,T}.
ii. Sample space for grade level in high school is {freshman, sophomore, junior, senior}.
IV.
Properties of Probability
a. P(A) = probability of event A
b. Complement of A: P( not A) = prob. of event A not happening
P( not A) = 1 – P(A)
c. P(A) + P( not A) = 1
d. Probability ranges from 0 to 1. 0  P( A)  1
e. P(A) = 0 means the event will never happen.
P(A) = 1 means it is guaranteed to happen
f. The sum of the probabilities in a sample space is 1.
P( entire sample space) = 1
V.
Conditional Probability
a. Conditional Probability is probability of an event A, given that event B has already taken place.
Another way is to ask for the probability of a particular event when the sample space is restricted to
only event B.
VI.
Dependent and Independent Events (Multiplication Rule) – “AND”
a. Independent Events – The first event does not affect the probability of the second event.
P(A and B) = P(A) * P(B)
b. Events A and B are independent if P(A) = P(A, given B).
c. Dependent Events – The first event affects the probability of the second event.
P( A and B) = P(A) * P(B, given A occurred)
VII.
Mutually Exclusive Events (Addition Rule) – “OR”
a. Mutually Exclusive Events – Two events can not occur at the same time.
P (A or B) = P(A) + P(B)
b. Not Mutually Exclusive Events – Two events can occur at the same time.
P(A or B) = P(A) +P(B) – P(A and B)
VIII.
Law of Large Numbers
In the long run, as the number of trials increases (gets larger and larger) the experimental probability
(relative frequency) gets closer and closer to the actual or theoretical probability.
IX.
Probability VS. Statistics
a. Probability deals with a known population.
b. Statistics deals with an unknown population or sample.
X.
Homework
a. Pg. 158 – 161 #1-3,8,10,12,15
b. Pg. 175 – 178 # 1-13 odd, 21
c. Pg. 194 – 196 #1, 3, 7,8, 13-20
Statistics – Chapter 4 – Probability – Examples
Probability
1. You are rolling a single die. Find the following.
a. Find the sample space.
b. P (2) =
c. P(7) =
d. P(not 5) =
e. P(even) =
f.
g.
h.
i.
P(not even) =
P(even) + P(not even) =
P(sample space) =
P(1) +P(2)+P(3)+P(4)+P(5)+P(6) =
Conditional Probability
2. You are rolling a single die. Find the following.
a. What is the probability of rolling a 6 after you have rolled a 2?
b. What is the probability of rolling an even number after you have rolled a 5?
c. What is the probability of rolling an odd number after you have rolled an even?
Dependent & Independent Events
3. Determine if the following events are dependent or independent.
Remember: Events A and B are independent if P(A) = P(A, given B).
a. Red card and Diamond
b. Red card and a Five
c. Spade and a Face Card
d. Six and a Face Card
Compound Events
4. You are rolling one die two times separately. Find the following.
a. Sample Space
b. P(1 and 2)=
c. P(even and even) =
5. Using a 52 card deck of cards, draw two cards (one at a time) without replacement, find the following
probabilities:
a. P(ace and king) =
b. P(2 and 2) =
c. P(jack, then queen, and then a king) =
6. Using a 52 card deck of cards, draw one card at a time, find the following probabilities:
a. What is the sample space?
d. P(ace or spade) =
b. P(ace or king) =
e. P(heart or less than 5) =
c. P (even or odd) =
f. P(queen of spades or any other spade) =
7. At Hopewell Electronics, all 140 employees were asked about their political affiliation. The employees were
grouped by type of work, as executives or production workers. The results with row and column totals are
shown in the table below.
Employee Type and Political Affiliation
Political Affiliation
Democrat (D)
Republican (R )
Independent (I) Row Total
Employee
Executive (E)
5
34
9
48
Production Worker (PW)
63
21
8
92
Column Total
68
55
17
140
a.
b.
c.
d.
e.
Compute P(D) and P(E)
Compute P(D, given E)
Are the events D and E independent?
Compute P (D and E)
Compute P (D or E)
a.
b.
c.
d.
Compute P(I) and P(PW)
Compute P(I, given PW)
Compute P(I and PW)
Compute P(I or PW)
8. Survey: Medical Tests – Diagnostic tests of medical conditions have several results. The test result can be
positive or negative, whether or not a patient has the condition ( + indicates a patient has the condition).
Consider a random sample of 200 patients, some of whom have a medical condition and some of whom do not.
Results of a new diagnostic test for the condition are shown. Assume the sample is representative of the entire
population. For a person to selected at random, compute the following probabilities:
a. P( +, given condition present)
Condition Condition
Row
b. P ( –, given condition present)
Present
Absent
Total
c. P (–, given condition absent)
Test Result
d. P (+, given condition absent)
+
110
20
130
e. P (condition present and +)
Test Result
f. P (condition present and – )
-20
50
70
Column
Total
130
70
200
Statistics – Chapter 4 – Probability – Examples – Answers
1a. {1,2,3,4,5,6}
1
1b. P(2)  .167
6
0
1c. P(7)   0
6
5
1d. P(not 5)  .833
6
3 1
1e. P(even)   .5
6 2
3 1
1f. P(noteven)   .5
6 2
1g. P(even)  P(noteven) 
3 3 6
  1
6 6 6
1h. P(sample space) = 1
1i. P(1) +P(2)+P(3)+P(4)+P(5)+P(6) =1
2a. {
(1,1), (!,2), (1,3), (1,4), (1,5), (1,6)
(2,1), (2,2), (2,3), (2,4), (2,5), 2(,6)
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
}
1 1 1
.028
2b. P( 1 and 2) = P(1)  P(2)   
6 6 36
3 3 9
.25
2c. P(even and even) = P( E )  P( E )   
6 6 36
1 1 1
.028
2d. P( 2 and 6) = P(2)  P(6)   
6 6 36
1 3 3
.083
2e. P(5 and even) = P(5)  P( E )   
6 6 36
3 3 9
.25
2f. P (even and odd) = P( E )  P(O)   
6 6 36
4 4
 .006
52 51
4 3
12
.005
3b. P(2 and 2) = P(2)  P(2, given2)   
52 51 2652
3a. P(ace and king) = P( A)  P( K , givenA) 
3c. P(jack, then queen, then a king) = P( J )  P(Q, givenJ )  P( K , givenJandQ) 
4 4 4
64
  
.0005
52 51 50 132600
4a. {
S1, S2, 3, S4, S5, S6, S7, S8, S9, S10, SJ, SQ, SK, SA
C1, C2, C3,C 4,C5, C6, C7, C8, C9, C10, J, CQ, CK, CA
H1, H2, H3, H4,H5, H6, H7,H8, H9, H10, HJ, HQ, HK, HA
D1, D2, D3, D4, D5,D 6,D 7, D8, D9, D10, DJ, DQ, DK, DA }
4
4
8


.154
4b. P(ace or king) = P( A)  P( K ) 
52 52 52
13 13 26


.5
4c. P(heart or spade) = P( H )  P( S ) 
52 52 52
20 16 36


.692
4d. P(even or odd) = P( E )  P(O) 
52 52 52
4 13
4 13 17 1 16
 [  ] 


.308
4e. P(ace or spade) = P( A)  P( S )  P( AandS ) 
52 52 52 52 52 52 52
4f. P(heart or less than 5) = P( H )  P( 5)  P( Hand  5) 
13 12 13 12
25 156
 [  ] 

.481.058 .423
52 52 52 52 52 2704
4h. P(even or diamond) = P( E )  P( D)  P( EandD) 
110
= .846
130
20
5b. P ( –, given condition present) =
= .154
130
50
.714
5c. P (–, given condition absent) =
70
20
.286
5d. P (+, given condition absent) =
70
110
.55
5e. P (condition present and +) =
200
20
.1
5f. P (condition present and – ) =
200
5a. P( +, given condition present) =
20 13 20 13 33 260
 [  ] 

.635.096 .539
52 52 52 52 52 2704