Download AP Statistics PowerPoint Notes Chapter 6 Probability

Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 6:
Probability and Simulation:
The Study of Randomness
Copyright © 2008 by W. H. Freeman & Company
Two computer simulations of tossing a balanced coin
100 times
A Head
A four
An Ace
Ex. What is the sample space for the roll of a die?
s= {1,2,3,4,5,6} All equal probability
What is the sample space for the roll of a pair of dice?
s={2,3,4,5,6,7,8,9,10,11,12} different probability
Events, Sample Spaces and Probability
• An event is a specific collection of sample
points:
– Event A: Observe an even number on the roll of a
single die. Often represented by a venn diagram.
• Tree diagrams help to determine the sample space
• Ex. An experiment consists of flipping a coin and
tossing a die.
• Ex. For your dinner you need to choose from six entrees,
eight sides, and five desserts. How many different combinations
of entrée, side and dessert are possible?
6x8x5 = 240
Unions and Intersections
Compound Events
Made of two or
more other events
Union
A B
Either A or B,
or both, occur
Intersection
A B
Both A and B
occur
Venn diagrams for (a) event (not E), (b) event (A & B),
and (c) event (A or B)
10
The Additive Rule and Mutually Exclusive Events
• Events A and B are mutually exclusive
(disjoint) if A  B contains no sample
points.
McClave: Statistics, 11th ed. Chapter 3:
Probability
12
Complementary Events
• The complement of any event A is the
event that A does not occur, AC.
A: {Toss an even number}
AC: {Toss an odd number}
B: {Toss a number ≤ 4}
BC: {Toss a number ≥ 5}
A  B = {1,2,3,4,6}
[A  B]C = {5}
(Neither A nor B occur)
Complementary Events
P ( A)  P ( A )  1
C
P ( A)  1  P ( A )
C
P ( A )  1  P ( A)
C
What is the probability of rolling a five when a pair of dice are
rolled?
P(A) = 4/36 = .111 =11.1%
• Independent Events – two events are independent if
the occurrence of one event does not change the
probability that the other occurs.
Ex. What is the probability of rolling a 7 each of three
consecutive roles of a pair of dice?
P(rolling 7) = 6/36 = 1/6 ; P( three 7s) = (1/6)3 =0.5%
If A and B are mutually exclusive,
P( A  B)  P( A)  P( B)
Ex.
Event A = { a household is prosperous; Inc. > 75 k}
Event B = { a household is educated; completed
college}
P(A) = 0.125
P(A and B) = 0.077
P(B) = 0.237
What is the probability the household is either
prosperous or educated?
P (A or B) = P(A) + P(B) – P(A and B) = 0.125 + 0.237 –
0.077
= 0.285
• Conditional probability - The probability of the occurrence
of an event given that another event has occurred.
♣ Notation – P(B|A) > probability of B given A
Ex.
A die has sides 1,2,3 painted red and sides 4,5,6
painted blue.
Event A = {Roll Blue}
Event B = {Roll Red}
Event C = {Roll even}
Event D = {Roll Odd}
What is the probability that you roll an even given
we rolled a blue?
What is the probability of rolling a red, even?
What is the probability that you roll an odd given
you rolled a blue?
• General Multiplication rule can be extended for any number
of events.
P(A and B and C) = P(A) x P(B|A) x P(C|A and B)
Example:
Deborah and Matthew have applied to become a partner in their law
firm. Over lunch one day they discuss the possibility. They
estimate that Deborah has a 70% chance of becoming partner
while Matthew has a 50% chance. They also believe that there
about a 30% chance that both of them will be chosen.
a)
What is the probability that either Deborah or Matthew become
partner?
b)
What is the probability that either Deborah or Matthew become
partner but not both?
c)
What is the probability that only Mathew becomes partner?
d)
What is the probability that neither one becomes partner?
e)
Are the events Independent?
a)
P(DUM) = P(D) + P(M) – P(D∩M)
= 0.7 + 0.5 – 0.3 = 0.9
b)
P(D∩Mc) U P(Dc∩M) = P(D∩Mc) + P(Dc∩M)
= 0.4 + 0.2 = 0.6
c)
P(Dc∩M) = P(M) x P(DcM) = 0.2
d)
P(Dc∩Mc) = 0.1
e)
P(D∩M) = P(D) x P(M) if independent
0.3 ≠ 0.7 x 0.5; No, they are not
independent
Event D = {Deborah is made partner}
Event M = {Matthew is made partner}
Example:
29% of internet users are 1-20 years old, 47% are 2150 years old and 24% are 51-90 years old. 47% of 120 year old users chat, 21% of 21-50 year old users
chat and 7% of 51-90 year old users chat
Event A1 = {person is 1-20 years old}
a)
What is the probability that a 19 year old internet user
does not chat?
Event C = {person chats online}
b)
What is the probability that an internet user chats?
c)
If we know an internet user chats, what is the
probability that they are 60 years old?
a) P(Cc ∩ A1) = P (A1) x P(Cc  A1) = 0.29 x 0.53 = 0.1537
b) P(C) = P(C ∩ A1) + P(C ∩ A2) + P(C ∩ A3)
0.1363 +0.0987 + 0.0168 = 0.2518
c) P(A3  C) = P(A3 ∩ C) / P(C) = 0.0168/(0.1363 +0.0987 +
0.0168)
= 0.0667
Event A2 = {person is 21-50 years old}
Event A3 = {person is 51-90 years old}