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Conditional Probability Conditional Probability

Conditional Probability

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Conditional Probability. P (A B) means. “the probability that event A occurs given that B has occurred”. Find ( i ) P (L) (ii) P (F ∩ L) (iii) P (F L). Ex 1. The following table gives data on the type of car, grouped by gas consumption, owned by 100 people. Low. Medium. High. - PowerPoint PPT Presentation

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Page 1: Conditional  Probability

Conditional Probability

Conditional Probability

Page 2: Conditional  Probability

Conditional Probability

“the probability that event A occurs given that B has occurred”.

P(A B) means

Page 3: Conditional  Probability

Conditional Probability

Female

LowMale

Ex 1. The following table gives data on the type of car, grouped by gas consumption, owned by 100 people.

42123

73312

HighMedium

One person is selected at random.L is the event “the person owns a low rated

car”F is the event “a female is chosen”.Find (i) P(L) (ii) P(F ∩ L) (iii) P(F L)

100

Total

Page 4: Conditional  Probability

Conditional Probability

2312

100100

(i) P(L) =

Solution:

Find (i) P(L) (ii) P(F ∩ L) (iii) P(F L)

35421Female733Male

TotalHighMediumLow

(leave the answer as fraction20 20

7

2312

735100

Low

Page 5: Conditional  Probability

Conditional Probability

(i) P(L) =

Solution:

Find (i) P(L) (ii) P(F and L) (iii) P(F L)

10042123Female73312Male

HighMediumLow

20 207

735

100

(ii) P(F ∩ L) = The probability of selecting a female with a low rated car.

23

23

100

Total

100

Page 6: Conditional  Probability

Conditional Probability

(i) P(L) =

Solution:

Find (i) P(L) (ii) P(F and L) (iii) P(F L)

1003542123Female73312Male

HighMediumLow

20 207

735

100

(ii) P(F ∩ L) =23

Total

100

(iii) P(F L)

The probability of selecting a female given the car is low rated.

23

23

35

12

Page 7: Conditional  Probability

Conditional Probability

(i) P(L) =

Solution:

Find (i) P(L) (ii) P(F and L) (iii) P(F L)

10042123Female73312Male

HighMediumLow

20 207

735

100

(ii) P(F ∩ L) =23

Total

100

(iii) P(F L)

2335

Page 8: Conditional  Probability

Conditional Probability

• If you don’t have a table, you can use this formula

( )( )( )

P A BP B AP A

Page 9: Conditional  Probability

Conditional Probability

Conditional ProbabilityResearchers asked people who exercise regularly whether they jog or

walk. Fifty-eight percent of the respondents were male. Twenty percent of all respondents were males who said they jog. Find the

probability that a male respondent jogs.

P( male ) = 58%P( male and jogs ) = 20%Let A = male.Let B = jogs.

The probability that a male respondent jogs is about 34%.

Write: P( A | B ) =P( A and B )P( A )

= Substitute 0.2 for P(A and B) and 0.58 for P(A). 0.344 Simplify.

0.2 0.58

Page 10: Conditional  Probability

Conditional Probability

Using Tree Diagrams Jim created the tree diagram  

after examining years of weatherobservations in his hometown. Thediagram shows the probability ofwhether a day will begin clear orcloudy, and then the probabilityof rain on days that begin clear

and cloudy.a. Find the probability that a day will start out clear, and then will rain.

The path containing clear and rain represents days that start out clear and then will rain.

P(clear and rain) = P(rain | clear) • P(clear)= 0.04 • 0.28

= 0.011The probability that a day will start out clear and then rain is about 1%.

Page 11: Conditional  Probability

Conditional Probability

Conditional Probability(continued)

b. Find the probability that it will not rain on any given day.

The paths containing clear and no rain and cloudy and no rain both represent a day when it will not rain. Find the probability for both paths and add them.

P(clear and no rain) + P(cloudy and no rain) =P(clear) • P(no rain | clear) + P(cloudy) • P(no rain | cloudy)

= 0.28(.96) + .72(.69)= 0.7656

The probability that it will not rain on any given day is about 77%.

Page 12: Conditional  Probability

Conditional Probability

Ex 3 In November, the probability of a man getting to work on time if there is fog on 285 is .

52

109If the visibility is good, the probability is .

203The probability of fog at the time he travels is .

(a)Calculate the probability of him arriving on time. (b)Calculate the probability that there was fog given that he arrives on time.

Page 13: Conditional  Probability

Conditional Probability

Not on time

52

2017

203

109

101

52

203

53

203

109

2017

101

2017

Fog

No Fog

On time

On timeNot on time

52

P(T F) 10

9P(T F/)

203

P(F)

53

F

F/

T

T/

T

T/

Page 14: Conditional  Probability

Conditional Probability

52

P(T F) 10

9P(T F/)

203

P(F)

52

2017

203

109

101

52

203

53

203

109

2017

101

2017

53

F

F/

T

T/

T

T/

Because we only reach the 2nd set of branches after the 1st set has occurred, the 2nd set must represent conditional probabilities.

Page 15: Conditional  Probability

Conditional Probability

(a)Calculate the probability of him arriving on time.

52

2017

203

109

101

52

203

53

203

109

2017

101

2017

53

F

F/

T

T/

T

T/

Page 16: Conditional  Probability

Conditional Probability

52

2017

203

109

101

53

203

109

2017

101

2017

53

F

F/

T

T/

T

T/

52

203

(a)Calculate the probability of him arriving on time.

1006

( foggy and he arrives on time )

Page 17: Conditional  Probability

Conditional Probability

52

2017

203

109

101

53

203

53

F

F/

T

T/

T

T/

109

2017

101

2017

52

203

1006

(a)Calculate the probability of him arriving on time.

200153

200153

1006

200165)(()( Tand/F Tand FT ) PPP

33

40 4033

( not foggy and he arrives on time )

Page 18: Conditional  Probability

Conditional Probability

Fog on 285 Getting to work

F

T

203

52

4033)( TPFrom part

(a),

)( Tand FP1006

52

203

(b)Calculate the probability that there was fog given that he arrives on time. We need

)( TFP

)()()(

T Tand F TF

PP

P

554( )P TF

)()((

T Tand F

T)FP

PP

4033

1006( ) TFP

3340

1006

5

22

11

Page 19: Conditional  Probability

Conditional ProbabilityExampl

e A sample of 100 adults were asked how they

travelled to work. The results are shown in the table. Walk Bus Bike Car

Men 10 12 6 26

Women 7 18 4 17

Total 100

Find (i) (ii) (iii) (iv)P(M)

P(M/ and C)

One person is picked at random.M is the event “the person is a man”C is the event “the person travels by car” P(M C) P(C M/)

Page 20: Conditional  Probability

Conditional Probability

100100

54

Total

174187Women

2661210Men

CarBikeBusWalk

Find (i) (ii) (iii) (iv)P(M)

P(M/ and C)

P(M C) P(C M/)

Solution:

(i) P(M)

(i)10054

5027

(ii) P(M C)

Page 21: Conditional  Probability

Conditional Probability

10043Total

174187Women

542661210Men

CarBikeBusWalk

Find (i) (ii) (iii) (iv)P(M)

P(M/ and C)

P(M C) P(C M/)

Solution:

(i) P(M) 10054

5027

(ii) P(M C)

(iii) P(M/ and C)

4326

Page 22: Conditional  Probability

Conditional Probability

10010043Total

174187Women

542661210Men

CarBikeBusWalk

Find (i) (ii) (iii) (iv)P(M)

P(M/ and C)

P(M C) P(C M/)

Solution:

(i) P(M) 10054

5027

(ii) P(M C)

(iii) P(M/ and C)

4326

10017 P(C M/) (iv)

Page 23: Conditional  Probability

Conditional Probability

10043Total

46174187Women

542661210Men

CarBikeBusWalk

Find (i) (ii) (iii) (iv)P(M)

P(M/ and C)

P(M C) P(C M/)

Solution:

(i) P(M) 10054

5027

(ii) P(M C)

(iii) P(M/ and C)

4326

10017 P(C M/) (iv)

4617

Page 24: Conditional  Probability

Conditional Probability

Independent Events

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

then A and B are independent events.

)()|( APBAP