Transcript NOTE - Parkway C-2

4-4 Multiplication Rules and
Conditional Probability
Objectives
-understand the difference between independent and dependent
events
-know how to use multiplication rule to calculate probability of
independent events.
-know how to use conditional probability to calculate probability of
dependent events.
4-4 The Multiplication Rules and
Conditional Probability
• Two events A and B are independent if
the fact that A occurs does not affect
the probability of B occurring.
• Example: Rolling a die and getting a 6,
and then rolling another die and getting
a 3 are independent events.
4-4 Multiplication Rules
When two events A and B
are independent , the
probability of both
occurring is
P ( A and B )  P ( A )  P ( B ).
4-4 Multiplication Rule 1 Example
• A card is drawn from a deck and
replaced; then a second card is drawn.
Find the probability of getting a queen
and then an ace.
• Solution: Because these two events are
independent (why?), P(queen and ace) =
(4/52)(4/52) = 16/2704 = 1/169.
4-4 Multiplication Rule 1 Example
• A Decima poll found that 46% of
Canadians say they suffer great stress at
least once a week. If three people are
selected at random, find the probability
that all three will say that they suffer stress
at least once a week.
• Solution: Let S denote stress. Then
P(S and S and S) = (0.46)3 = 0.097.
4-4 Multiplication Rule 1 Example
• The probability that a specific medical test
will show positive is 0.32. If four people
are tested, find the probability that all four
will show positive.
• Solution: Let T denote a positive test
result. Then P(T and T and T and T) =
(0.32)4 = 0.010.
Multiplication Rules:
Conditional Probability
• When the outcome or occurrence of the
first event affects the outcome or
occurrence of the second event in such a
way that the probability is changed, the
events are said to be dependent.
• Example: Having high grades and getting
a scholarship are dependent events.
Multiplication Rules:
Conditional Probability
• The conditional probability of an event B in
relationship to an event A is the probability
that an event B occurs after event A has
already occurred.
• The notation for the conditional probability of
B given A is P(B|A).
• NOTE: This does not mean B  A.
4-4 Multiplication Rule 2
When two events A and B
are dependent , the
probability of both
occurring is
P ( A and B )  P ( A)  P ( B| A).
The Multiplication Rules:
Conditional Probability - Example
• In a shipment of 25 microwave ovens, two
are defective. If two ovens are randomly
selected and tested, find the probability
that both are defective if the first one is not
replaced after it has been tested.
• Solution: See next slide.
The Multiplication Rules and
Conditional Probability - Example
• Solution: Since the events are
dependent, P(D1 and D2)
= P(D1)P(D2| D1) = (2/25)(1/24)
= 2/600 = 1/300.
The Multiplication Rules and
Conditional Probability - Example
• The KW Insurance Company found that 53%
of the residents of a city had homeowner’s
insurance with its company. Of these clients,
27% also had automobile insurance with the
company. If a resident is selected at random,
find the probability that the resident has both
homeowner’s and automobile insurance.
The Multiplication Rules and
Conditional Probability - Example
• Solution: Since the events are
dependent, P(H and A)
= P(H)P(A|H) = (0.53)(0.27)
= 0.1431.
The Multiplication Rules and
Conditional Probability - Example
• Box 1 contains two red marbles and one
blue marble. Box 2 contains three blue
marbles and one red marble. A coin is
tossed. If it falls heads up, box 1 is
selected and a marble is drawn. If it falls
tails up, box 2 is selected and a marble is
drawn. Find the probability of selecting a
red marble.
Tree Diagram for Example
P(R|B1) 2/3
P(B1) 1/2
P(B2) 1/2
Red (1/2)(2/3)
Box 1
Blue (1/2)(1/3)
P(B|B1) 1/3
P(R|B2) 1/4
Box 2
Red (1/2)(1/4)
P(B|B2) 3/4 Blue (1/2)(3/4)
The Multiplication Rules and
Conditional Probability - Example
• Solution: P(red) = (1/2)(2/3) +
(1/2)(1/4) = 2/6 + 1/8 = 8/24 + 3/24
= 11/24.
Conditional Probability Formula
The probability that the second event B occurs
given that the first event A has occurred can be
found by dividing the probability that both events
occurred by the probability that the first event has
occurred . The formula is
P( A and B)
.
P( B| A) =
P( A)
Conditional Probability Example
• The probability that Sam parks in a noparking zone and gets a parking ticket is 0.06,
and the probability that Sam cannot find a
legal parking space and has to park in the noparking zone is 0.2. On Tuesday, Sam
arrives at school and has to park in a noparking zone. Find the probability that he will
get a ticket.
Conditional Probability Example
• Solution: Let N = parking in a noparking zone and T = getting a ticket.
• Then P(T|N) = [P(N and T) ]/P(N) =
0.06/0.2 = 0.30.
Conditional Probability Example
• A recent survey asked 100 people if
they thought women in the armed
forces should be permitted to
participate in combat. The results are
shown in the table on the next slide.
Conditional Probability Example
Gender
Gender
Yes
Yes
No
No
Total
Total
Male
Male
32
32
18
18
50
50
Female
Female
88
42
42
50
50
Total
Total
40
40
60
60
100
100
Conditional Probability Example
• Find the probability that the respondent
answered “yes” given that the respondent
was a female.
• Solution: Let M = respondent was a male; F
= respondent was a female;
Y=
respondent answered “yes”;
N=
respondent answered “no”.
Conditional Probability Example
• P(Y|F) = [P( F and Y) ]/P(F) = [8/100]/[50/100]
= 4/25.
• Find the probability that the respondent was a
male, given that the respondent answered
“no”.
• Solution: P(M|N) = [P(N and M)]/P(N) =
[18/100]/[60/100] = 3/10.
Homework
• Pg. 215 # 1-49 every other odd