Review Probabilities –Definitions of experiment, event, simple event, sample space, probabilities, intersection, union compliment –Finding Probabilities

Review

Probabilities– Definitions of experiment, event, simple event,

sample space, probabilities, intersection, union compliment

– Finding Probabilities– Drawing Venn Diagrams – If A and B are two events then

P(A or B) = P(A) + P(B) - P(A and B),

P(not A) = 1 - P(A). – Two events A and B are mutually exclusive if P(A and

B) = 0.

Example

Suppose P (E) = 0.4, P (F) = 0.3, and

P(E or F)=0.6. Find:

P(E and F) P(EC or FC)

P(EC and FC) P(EC and F)

Example

Suppose P (E) = 0.4, P (F) = 0.3, and


P(E and F) =0.1 P(EC or FC)


Example

Suppose P (E) = 0.4, P (F) = 0.3, and


P(E and F) =0.1 P(EC or FC)=0.9


Example

Suppose P (E) = 0.4, P (F) = 0.3, and



P(EC and FC)=0.4 P(EC and F)

Example

Suppose P (E) = 0.4, P (F) = 0.3, and



P(EC and FC)=0.4 P(EC and F)= 0.2

7

Problems

• 3.15, 3.20, 3.44, 3.45, 3.54

3.15

2 1 2 3 1 3 2 2 1 2

2 1 1 1 3 2 3 1 2 1

BB ,BR ,BR ,BR ,BR

,BR ,BR ,RR ,RR ,RR a)

3.15

2 1 2 3 1 3 2 2 1 2

2 1 1 1 3 2 3 1 2 1

BB ,BR ,BR ,BR ,BR


each 10

1 b)

3.15

2 1 2 3 1 3 2 2 1 2

2 1 1 1 3 2 3 1 2 1

BB ,BR ,BR ,BR ,BR


each 10

1 b)

; 10

3;

10

6;

10

1 c)

3.20

Rhino. Whiteand

RhinoBlack :pickyou rhino theofcolour The a)

3.20

Rhino. Whiteand


)RhinoBlack (

)Rhino White(

s.frequencie relative Useb)

P

P

3.20

Rhino. Whiteand


764.84002600

8400)RhinoBlack (

236.84002600

2600)Rhino White(

s.frequencie relative Useb)

P

P

3.44

up. added

iesprobabilit with theDiagramVenn theDraw

3.44

up. added


.69 f) .31 e) .97 d) .5 c) .19 b) .53 a)

3.44

up. added


.69 f) .31 e) .97 d) .5 c) .19 b) .53 a)

empty.not

ison intersecti therebecausenot are they No g)

3.45

3.45

Low Medium High

On 0.50 0.10 0.05

Off 0.25 0.07 0.03

3.45

Low Medium High Total

On 0.50 0.10 0.05 0.65

Off 0.25 0.07 0.03 0.35

Total 0.75 0.17 0.08 1

3.45


On 0.50 0.10 0.05 0.65

Off 0.25 0.07 0.03 0.35

Total 0.75 0.17 0.08 1

}High{D Low}, and {OffC On},or {Med.B {On},A

3.45


On 0.50 0.10 0.05 0.65

Off 0.25 0.07 0.03 0.35

Total 0.75 0.17 0.08 1


B) andP(A , B)or P(A , )P(A

P(D), P(C), P(B) , P(A)C

3.45


On 0.50 0.10 0.05 0.65

Off 0.25 0.07 0.03 0.35

Total 0.75 0.17 0.08 1


65.0 B) andP(A , P(B)) (see 0.72 B)or P(A

0.350.65-1P(A)-1 )P(A

0.08 P(D), 25.0P(C)

0.720.10-0.650.17 P(B) , 0.65 P(A)

C

3.54

3.54

on.intersecti no haveThey YES. g)

intersectThey NO. f)

.843 e)

0.64476/6488 d)

.569

787/7488670/74882806/74883) P(below c)

.1921438/7488 weapon)P(carried b)

175.7488/1307P(5) a)

Conditional Probability

You are dealt two cards from a deck. What is the probability the first card dealt is a Jack?


You are dealt two cards from a deck. What is the probability the first card dealt is a Jack?

13

1:Answer


You are dealt two cards from a deck. What is the probability of the second card was a jack given the first card was not a jack is called card dealt is a Jack?





51

4:Answer



The probability of drawing jack given the first card was not a jack is called conditional probability. A key words to look for is “given that.”


The probability of drawing jack given the first card was not a jack is called conditional probability. A key words to look for is “given that.”

The probability that the event A occurs, given that B occurs is denoted:

This is read the probability of A given B.).|( BAP


You are dealt two cards from a deck. What is the probability the second card dealt is a Jack?





13

1:Answer

Example

A local union has 8 members, 2 of whom are women. Two are chosen by a lottery to represent the union.

Example


a) What is the probability they are both male?

Example



M)|P(M(M):Answer P

Example



536.07

5

8

6:Answer

Example


b) What is the probability they are both female?

Example


b) What is the probability they are both female?

036.07

1

8

2F)|P(F*P(F):Answer


How would we draw the event A given B?

A B A

and B



Since we know B has occurred, we ignore everything else.

A B A

and B




A B A

and B

Conditional ProbabilityHow would we draw the event A given B?


With some thought this tells us:

B A

and B

)(

) and ()|(

BP

BAPBAP

Conditional ProbabilitySince we know B has occurred, we ignore everything else.

Or rearranging:

B A

and B

)|()() and ( BAPBPBAP

Example

Find the probability of selecting an all male jury from a group of 30 jurors, 21 of whom are men.

Example

Find the probability of selecting an all male jury from a group of 30 jurors, 21 of whom are men.

Solution: P(12 M) =P(M)*P(M|M)*P(M|MM) * ….

= 21/30 * 20/29 * 19/28 * 18/27 * … 10/19

= 0.00340

Independent Events

Two events A and B are independent if the occurrence of one does not affect the probability of the other.

Independent Events


Two events A and B are independent then P(A|B) = P(A).

Independent Events


Two events A and B are independent then P(A|B) = P(A).

Two events which are not independent are dependent.

Multiplication Rule

Multiplication Rule:

For any pair of events:

P(A and B) = P(A) * P(B|A)

Multiplication Rule

Multiplication Rule:

For any pair of events:


For any pair of independent events:

P(A and B) = P(A) * P(B)

Multiplication RuleFor any pair of events:


For any pair of independent events:

P(A and B) = P(A) * P(B)

If P(A and B) = P(A) * P(B), then A and B are independent.

Multiplication RuleMultiplication Rule:

P(A and B) = P(A) * P(B) if A and B are independent.

P(A and B) = P(B) * P(A|B) if A and B are dependent.

Note: The multiplication rule extends to several events: P(A and B and C) =P(C)*P(B|C)*P(A|BC)

ExampleA study of 24 mice has classified the mice by two categories

Black White Grey

Eye Colour

Red Eyes 3 5 2

Black Eyes 1 7 6

Fur Colour

A study of 24 mice has classified the mice by two categories

a) What is the probability that a randomly selected mouse has white fur?

b) What is the probability it has black eyes given that it has black fur?

c) Find pairs of mutually exclusive and independent events.

Black White Grey

Eye Colour

Red Eyes 3 5 2

Black Eyes 1 7 6

Fur Colour


a) What is the probability that a randomly selected mouse has white fur? 12/24=0.5

b) What is the probability it has black eyes given that it has black fur?


Black White Grey

Eye Colour

Red Eyes 3 5 2

Black Eyes 1 7 6

Fur Colour


a) What is the probability that a randomly selected mouse has white fur? 12/24=0.5

b) What is the probability it has black eyes given that it has black fur? 1/4=0.25


Black White Grey

Eye Colour

Red Eyes 3 5 2

Black Eyes 1 7 6

Fur Colour

b) What is the probability it has black eyes given that it has black fur? 1/4=0.25


ME: Black eyes and Red eyes

IND: White Fur and Red Eyes; Black Fur and Red Eyes

Black White Grey

Eye Colour

Red Eyes 3 5 2

Black Eyes 1 7 6

Fur Colour

Descriptive Phrases

Descriptive Phrases require special care!

– At most– At least– No more than– No less than

Review

• Conditional Probabilities

• Independent events

• Multiplication Rule

• Tree Diagrams

62

Homework

• Review Chapter 3

• Read Chapter 4.1-4.4

• Quiz on Tuesday on Chapters 1 and 2

• Problems on next slide

Problems

Problems 3.66, 3.68, 3.70, 3.75, 3.80, 3.87

Documents

Review Probabilities –Definitions of experiment, event, simple event, sample space, probabilities, intersection, union compliment –Finding Probabilities