Download Math 115a * Section 2

Conditional Probability:
Independent Events
Finite 7-5
example:
• Toss a balanced die once and record the number
on the top face.
• Let E be the event that a 1 shows on the top face.
• Let F be the event that the number on the top face
is odd.
– What is P(E)?
– What is the Probability of the event E if we are told that the
number on the top face is odd, that is, we know that the
event F has occurred?
Conditional Probability
• Key idea: The original sample space no
longer applies.
• The new or reduced sample space is
S={1, 3, 5}
• Notice that the new sample space
consists only of the outcomes in F.
• P(E occurs given that F occurs) = 1/3
• Notation: P(E|F) = 1/3
Conditional Probability
• Def. The conditional probability of E given F
is the probability that an event, E, will occur
given that another event, F, has occurred
P( E  F )
P( E | F ) 
P( F )
if
P( F )  0
Conditional Probability
P( A  B)
P( A B) 
P( B)
A
B
S
Conditional Probability
• If the outcomes of an experiment are equally
likely, then
number of outcomes in E  F
P( E | F ) 
number of outcomes in F
Conditional Probability
• Example:
Earned degrees in the United States in
recent year
Female
Male
Total
B
616
529
1145
M
194
171
365
P
30
44
74
D
16
26
42
Total
856
770
1626
529
P( Male | B) 
 0.4620
1145
770
P( Male ) 
 0.4735
1626
Conditional Probability
The academy awards is soon to be shown.
For a specific married couple the probability
that the husband watches the show is 80%,
the probability that his wife watches the
show is 65%, while the probability that they
both watch the show is 60%.
If the husband is watching the show, what is
the probability that his wife is also watching
the show
Example
The academy awards is soon to be shown.
Let B = the event that the husband watches the
show
P[B]= 0.80
Let A = the event that his wife watches the show
P[A]= 0.65 and P[A ∩ B]= 0.60
P  A B  
P  A  B
P  B
0.60

 0.75
0.80
Example
P( E  F )
P( E | F ) 
P( F )
Conditional Probability can be rewritten as follows
P( E  F )  P( E | F ) * P( F )
Conditional Probability
Example:
E: dollar falls in value against the yen
F: supplier demands renegotiation of contract
P( E )  0.40
P( F | E )  0.8
Find
P( E  F )
P( E  F )  0.8 * 0.4  0.32
Conditional Probability
mutually exclusive
Two mutually exclusive events are independent only in
the special case where
P  A  0 and P  B  0. (also P  A  B  0
A
B
Mutually exclusive events are
highly dependent otherwise. A
and B cannot occur
simultaneously. If one event
occurs the other event does not
occur.
Difference between independence
and mutually exclusive
P  A  B  P  A P  B
or
S
P  A  B
P  B
 P  A 
P  A
P S 
B
A
A B
The ratio of the probability of the
set A within B is the same as the
ratio of the probability of the set
A within the entire sample S.
Difference between independence
and mutually exclusive

 P  A P  B A if P  A  0
P  A  B  

 P  B  P  A B  if P  B   0
and
P  A  B  P  A P  B
if A and B are independent.
The multiplicative rule of probability
If the probability of the occurrence of event A
is the same regardless of whether or not an
outcome B occurs, then the outcomes A and
B are said to be independent of one another.
Symbolically, if
P( A | B)  P( A)
then A and B are independent events.
Independent Events
P( A  B)  P( A | B) P( B)
then we can also state the following
relationship for independent events:
P( A  B)  P( A) P( B)
if and only if
A and B are independent events.
Independent Events
• A coin is tossed and a single 6-sided die is
rolled. Find the probability of getting a
head on the coin and a 3 on the die.
• Probabilities:
P(head) = 1/2
P(3) = 1/6
P(head and 3) = 1/2 * 1/6 = 1/12
Example
• Example:
If E, F, and G are independent, then
P( E  F  G)  P( E ) * P( F ) * P(G)
Independence Formula –3 events
• If E, F, and G are independent given that
an event H has occurred, then
P( E  F  G | H )  P( E | H ) * P( F | H ) * P(G | H )
The Notion of Independence applied to
Conditional Probability
• Independent Events vs. Mutually Exclusive Events
(Disjoint Events)
• If two events are Independent,
P( A | B)  P( A)
P( A  B)  P( A) P( B)
• If two events are Mutually Exclusive Events then
they do not share common outcomes
Important
Summary of the Rules of
Probability
P[A  B] = P[A] + P[B] – P[A  B]
and
P[A  B] = P[A] + P[B] if P[A  B] = f
The Additive Rule
for any event E
P  E   1  P  E 
The Rule for Complements
P  A B  
P  A  B
P  B
Conditional probability

 P  A P  B A if P  A  0
P  A  B  


P
B
P
A
B
if
P
B

0








and
P  A  B  P  A P  B
if A and B are independent.
This is the definition of independent
The multiplicative rule of probability
• you have a box with 3 blue marbles, 2 red
marbles, and 4 yellow marbles. You are
going to pull out one marble, record its color,
put it back in the box and draw another
marble. What is the probability of pulling out
a red marble followed by a blue marble?
Example
• you have a box with 3 blue marbles, 2 red marbles, and 4
yellow marbles. You are going to pull out one marble, record
its color, put it back in the box and draw another marble.
What is the probability of pulling out a red marble followed
by a blue marble?
Example
• Suppose you are going to draw two cards
from a standard deck. What is
the probability that the first card is an ace
and the second card is a jack (just one of
several ways to get “blackjack” or 21).
Example
• Suppose you are going to draw two cards
from a standard deck. What is
the probability that the first card is an ace
and the second card is a jack (just one of
several ways to get “blackjack” or 21).
Example
• Pages 330 – 333
• 1-15 odd, 21, 23, 29, 33 – 43 odd, 49 – 53
all, 64 – 67 all, 73
Homework