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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
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
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? White workers are slightly more likely to be paid at or below minimal wage than workers in general.
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
P(at or Below) =1,94671,966
=0.0270
P(at or Below |White) =1,68159,877
=0.0281
Section 5.4 - Conditional Probability
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.
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?
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
Section 5.4 - Conditional Probability
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.
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.
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
P(Black) =9,41771,966
=0.1309
P(Black | at or Below) =2271,946
=0.1166
Section 5.4 - Conditional Probability
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)
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
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.
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
Write a formula that tells how the three probabilities in part b are related. Compare it to the computation on p 328.
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
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?
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
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)
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
P30. Use the Multiplication Rule to find the probability of getting a sum of 8 and doubles when you roll two dice.
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
P31. Suppose you roll two dice. Use the definition of conditional probability to find P(D|8). Compare this probability with P(8|D).
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
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.
Section 5.4 - Conditional Probability
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
Section 5.4 - Conditional Probability
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?
Section 5.4 - Conditional Probability
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%