4-4 Multiplication Rules and Conditional Probability Objectives -understand the difference between...

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.

• Two events A and B are independentindependent if the fact that A occurs does not affect the probability of B occurring.

• Example:Example: Rolling a die and getting a 6, and then rolling another die and getting a 3 are independent events.

4-4 The Multiplication Rules and 4-4 The Multiplication Rules and Conditional Probability Conditional Probability

4-4 Multiplication Rules4-4 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 -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:Solution: Because these two events are independent (why?), P(queen and ace) = (4/52)(4/52) = 16/2704 = 1/169.


• 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:Solution: Let S denote stress. Then P(S and S and S) = (0.46)3 = 0.097.


• 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:Solution: Let T denote a positive test result. Then P(T and T and T and T) = (0.32)4 = 0.010.

• 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:Example: Having high grades and getting a scholarship are dependent events.

Multiplication Rules: Multiplication Rules: Conditional Probability Conditional Probability

• The conditional probabilityconditional 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:NOTE: This does not mean B A.

Multiplication Rules: Multiplication Rules: Conditional Probability Conditional Probability

4-4 Multiplication Rule 24-4 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

,

( ) ( ) ( | ).

• 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:Solution: See next slide.

The Multiplication Rules: The Multiplication Rules: Conditional Probability - Conditional Probability - Example

• Solution: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 The Multiplication Rules and Conditional Probability - 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.


• Solution:Solution: Since the events are dependent, P(H and A) = P(H)P(A|H) = (0.53)(0.27) = 0.1431.


• 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 Tree Diagram for Example

P(B1) 1/2

Red

Red

Blue

Blue

Box 1

P(B2) 1/2Box 2

P(R|B1) 2/3

P(B|B1) 1/3

P(R|B2) 1/4

P(B|B2) 3/4

(1/2)(2/3)

(1/2)(1/3)

(1/2)(1/4)

(1/2)(3/4)

• Solution:Solution: P(red) = (1/2)(2/3) + (1/2)(1/4) = 2/6 + 1/8 = 8/24 + 3/24 = 11/24.


Conditional Probability -Conditional Probability - FormulaFormula

.

( | ) =( )

( )

The probability that the 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 B AP A and B

P A

second

.

• The probability that Sam parks in a no-parking 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 no-parking zone is 0.2. On Tuesday, Sam arrives at school and has to park in a no-parking zone. Find the probability that he will get a ticket.

Conditional Probability -Conditional Probability - Example

• Solution:Solution: Let N = parking in a no-parking zone and T = getting a ticket.

• Then P(T |N) = [P(N and T) ]/P(N) = 0.06/0.2 = 0.30.


• 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.



Gender Yes No Total

Male 32 18 50

Female 8 42 50

Total 40 60 100

Gender Yes No Total

Male 32 18 50

Female 8 42 50

Total 40 60 100

• Find the probability that the respondent answered “yes” given that the respondent was a female.

• Solution:Solution: Let M = respondent was a male; F = respondent was a female; Y = respondent answered “yes”; N = respondent answered “no”.


• 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

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