section 5.4 - conditional probability p23. for the titanic data in display 5.39, let s be the event...

Section 5.4 - Conditional Probability

P23. For the Titanic data in Display 5.39, let S be the event a person survived and F be the event a person was female. Find and interpret these probabilities.

a. P(F)

b. P(F|S)

c. P(not F)

d. P(not F|S)

e. P(S|not F)

Display 5.39Gender

Male Female Total

Survived?

Yes 367 344 711

No 1364 126 1490

Total 1731 470 2201


P23.

P(F) =4702201

=0.2135 P(F |S) =344711

=0.4838

P(not F ) =17312201

=0.7865 P(not F |S) =367711

=0.5162

P(S|not F ) =3671731

=0.2120

Display 5.39Gender

Male Female Total

Survived?

Yes 367 344 711

No 1364 126 1490

Total 1731 470 2201


P24. Display 5.44 gives the hourly workers in the U.S., classified by race and by whether they were paid at or below minimum wage or above minimum wage. You select an hourly worker at random.

a. Find P(at or Below)

b. Find P(at or Below|White)

c. What does a comparison of the two probabilities in parts a and b tell you?

RacePaid at or Below Minimum Wage

Paid Above Minimum Wage

Total

White 1,681 58,196 59,877

Black 227 9,190 9,417

Asian 38 2,634 2,672

Total 1,946 70,020 71,966



a. Find P(at or Below)

b. Find P(at or Below|White)

c. What does a comparison of the two probabilities in parts a and b tell you? White workers are slightly more likely to be paid at or below minimal wage than workers in general.



Total

White 1,681 58,196 59,877

Black 227 9,190 9,417

Asian 38 2,634 2,672

Total 1,946 70,020 71,966

P(at or Below) =1,94671,966

=0.0270

P(at or Below |White) =1,68159,877

=0.0281



d. Find P(Black)

e. Find P(Black|at or Below)

f. What does a comparison of the two probabilities in parts d and e tell you?



Total

White 1,681 58,196 59,877

Black 227 9,190 9,417

Asian 38 2,634 2,672

Total 1,946 70,020 71,966



d. Find P(Black)

e. Find P(Black|at or Below)

f. What does a comparison of the two probabilities in parts d and e tell you? A worker who is paid at or below minimum wage is less likely to be black than a worker selected at random.



Total

White 1,681 58,196 59,877

Black 227 9,190 9,417

Asian 38 2,634 2,672

Total 1,946 70,020 71,966

P(Black) =9,41771,966

=0.1309

P(Black | at or Below) =2271,946

=0.1166


P25.Suppose Jack draws marbles at random, without replacement, from a bag containing three red and two blue marbles. Find these conditional probabilities.

a. P(2nd is red|1st is red)

b. P(2nd is red|1st is blue)

c. P(3rd is blue|1st is red and 2nd is blue)

d. P(3rd is red|1st is red and 2nd is red)


P25.Suppose Jack draws marbles at random, without replacement, from a bag containing three red and two blue marbles. Find these conditional probabilities.

{R,R,R,B,B}

P(2nd is red | 1st is red) =24=12=0.5000

P(2nd is red|1st is blue) =34=0.7500

P(3rd is blue|1st is red and 2nd is blue) =13=0.3333

P(3rd is red|1st is red and 2nd is red) =13=0.3333


P26. Suppose Jill draws a card from a standard 52-card deck. Find the probability that

a. It is a club, given that it is black.

b. It is a jack, given that it is a heart.

c. It is a heart, given that it is a jack.


P26. Suppose Jill draws a card from a standard 52-card deck. Find the probability that

a. It is a club, given that it is black.

b. It is a jack, given that it is a heart.

c. It is a heart, given that it is a jack.

P(Club | Black) =1326

=12=0.5000

P(Jack | Heart) =113

=0.0769

P(Heart | Jack) =14=0.2500


P27. Look again at the Titanic data in Display 5.39.

Make a tree diagram to illustrate this situation, this time branching first on whether the person survived.

Write these probabilities as unreduced fractions.

P(S); P(F|S); P(S and F)

Display 5.39Gender

Male Female Total

Survived?

Yes 367 344 711

No 1364 126 1490

Total 1731 470 2201


Make a tree diagram to illustrate this situation, this time branching first on whether the person survived.

P(F and S) = 344 / 2201P(M and S) = 367 / 2201

P(F and D) = 126 / 2201P(M and D) = 1364 / 2201P(M|D) = 1364 / 1490

Died

Male Male and Died

Female Female and Died

P(F|D) = 126/ 1490

P(D)= 1490 / 2201

P(M|S) = 367 / 711

Male Male and SurvivedP(S)= 711 / 2201

Female Female and Survived

SurvivedP(F|S) = 344 / 711


Write a formula that tells how the three probabilities in part b are related. Compare it to the computation on p 328.


Write a formula that tells how the three probabilities in part b are related. Compare it to the computation on p 328.

P(S) =7112201

; P(F |S) =344711

; P(S∩ F ) =3442201

P(S)⋅P(F |S) =P(S∩ F )7112201

⋅344711

=3442201


P28. Use the Multiplication Rule to find the probability that if you draw two cards from a deck without replacing the first before drawing the second, both cards will be hearts. What is the probability if you replace the first card before drawing the replacement?


P28. Use the Multiplication Rule to find the probability that if you draw two cards from a deck without replacing the first before drawing the second, both cards will be hearts. What is the probability if you replace the first card before drawing the replacement?

Without replacement : P(HH ) =1352

⋅1251

=117

=0.0588

With replacement : P(HH ) =1352

⋅1352

=116

=0.0625


P29. Suppose you take a random sample of size n = 2, without replacement, from the population {W,W,M,M}. Find these probabilities:

P(W chosen 1st)

P(W chosen 2nd|W chosen 1st)

P(W chosen 1st and W chosen 2nd)


P29. Suppose you take a random sample of size n = 2, without replacement, from the population {W,W,M,M}. Find these probabilities:

P(W chosen 1st) =24=12=0.5000

P(W chosen 2nd|W chosen 1st) =13=0.3333

P(W chosen 1st and W chosen 2nd) =24⋅13=16=0.1667


P30. Use the Multiplication Rule to find the probability of getting a sum of 8 and doubles when you roll two dice.


P30. Use the Multiplication Rule to find the probability of getting a sum of 8 and doubles when you roll two dice.

P(8 ∩D) =P(8)⋅P(D |8) =536

⋅15=

136

=0.0278

This is not the same as

P(8)⋅P(D) =536

⋅636

=5

216=0.0231


P31. Suppose you roll two dice. Use the definition of conditional probability to find P(D|8). Compare this probability with P(8|D).


P31. Suppose you roll two dice. Use the definition of conditional probability to find P(D|8). Compare this probability with P(8|D).

P(D | 8) =P(D∩8)

P(8)=

136536

=15=0.2000

P(8 |D) =P(D∩8)

P(D)=

136636

=16=0.1667


P32. Suppose you know that, in a class of 30 students, 10 students have blue eyes and 20 students have brown eyes. Twenty-four of the students are right-handed, and 6 are left-handed. Of the left-handers, 2 have blue eyes. Make a two-way table showing this situation. Then use the definition of conditional probability to find the probability that a student randomly selected from this class is right-handed, given that the student has brown eyes.


P32. Suppose you know that, in a class of 30 students, 10 students have blue eyes and 20 students have brown eyes. Twenty-four of the students are right-handed, and 6 are left-handed. Of the left-handers, 2 have blue eyes. Make a two-way table showing this situation. Then use the definition of conditional probability to find the probability that a student randomly selected from this class is right-handed, given that the student has brown eyes.

Right-Handed Left-Handed Total

Blue Eyes 8 2 10

Brown Eyes 16 4 20

Total 24 6 30

P(R | Br) =P(R∩Br)

P(Br)=1630

÷2030

=1620

=0.8000


P33. As of July 1 of a recent season, the L.A. Dodgers had won 53% of their games. 18% of their games had been played against left-handed starting pitchers. The Dodgers won 36% of the games played against left-handed starting pitchers. What percentage of their games against right-handed starting pitchers did they win?


P33. As of July 1 of a recent season, the L.A. Dodgers had won 53% of their games. 18% of their games had been played against left-handed starting pitchers. The Dodgers won 36% of the games played against left-handed starting pitchers. What percentage of their games against right-handed starting pitchers did they win?

For a typical set of 100 games, the W-L record would look approximately like the following table:

Right-Handed Pitcher Left-Handed Pitcher Total

Won 46.52 6.48 = .36 x 18 53

Lost 35.48 11.52 47

Total 82 18 100

percentage against RH =46.5282

=56.7%

section 5.4 - conditional probability p23. for the titanic data in display 5.39, let s be the event...

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