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8.4.1: Conditional Probability (Exercises) - Mathematics


SECTION 8.4 PROBLEM SET: CONDITIONAL PROBABILITY

Questions 1 - 4: Do these problems using the conditional probability formula: (P(A | B)=frac{P(A cap B)}{P(B)}).

  1. A card is drawn from a deck. Find the conditional probability of (P)(a queen | a face card).
  1. A card is drawn from a deck. Find the conditional probability of (P)(a queen | a club).
  1. A die is rolled. Find the conditional probability that it shows a three if it is known that an odd number has shown.
  1. If (P(A)) = .3 , (P(B)) = .4, (P)((A) and (B)) = .12, find:
    1. (P(A | B))
    2. (P(B | A))

Questions 5 - 8 refer to the following: The table shows the distribution of Democratic and Republican U.S. Senators by gender in the 114th Congress as of January 2015.

MALE(M)

FEMALE(F)

TOTAL

DEMOCRATS (D)

30

14

44

REPUBLICANS(R)

48

6

54

OTHER (T)

2

0

2

TOTALS

80

20

100

Use this table to determine the following probabilities:

  1. (P(M | D))
  1. (P(D | M))
  1. (P(F | R))
  1. (P(R | F))

Do the following conditional probability problems.

  1. At a college, 20% of the students take Finite Math, 30% take History, and 5% take both Finite Math and History. If a student is chosen at random, find the following conditional probabilities.
    1. He is taking Finite Math given that he is taking History.
    2. He is taking History assuming that he is taking Finite Math.
  1. At a college, 60% of the students pass Accounting, 70% pass English, and 30% pass both of these courses. If a student is selected at random, find the following conditional probabilities.
    1. He passes Accounting given that he passed English.
    2. He passes English assuming that he passed Accounting.
  1. If (P(F) = .4), (P(E | F) = .3), find (P)((E) and (F)).
  1. (P(E) = .3), (P(F) = .3); (E) and (F) are mutually exclusive. Find (P(E | F)).
  1. If (P(E) = .6), (P)((E) and (F)) = .24, find (P(F | E)).
  1. If (P)((E) and (F)) = (.04), (P(E | F) = .1), find (P(F)).

At a college, 72% of courses have final exams and 46% of courses require research papers. 32% of courses have both a research paper and a final exam. Let (F) be the event that a course has a final exam and (R) be the event that a course requires a research paper.

  1. Find the probability that a course has a final exam given that it has a research paper.
  1. Find the probability that a course has a research paper if it has a final exam.

SECTION 8.4 PROBLEM SET: CONDITIONAL PROBABILITY

Consider a family of three children. Find the following probabilities.

  1. (P)(two boys | first born is a boy)
  1. (P)(all girls | at least one girl is born)
  1. (P)(children of both sexes | first born is a boy)
  1. (P)(all boys | there are children of both sexes)

Questions 21 - 26 refer to the following:
The table shows highest attained educational status for a sample of US residents age 25 or over:

(D) Did not Complete

High School

(H) High School

Graduate

(C)

Some

College

(A) Associate

Degree

(B) Bachelor

Degree

(G)

Graduate

Degree

TOTAL

25-44 (R)

95

228

143

81

188

61

796

45-64 (S)

83

256

136

80

150

67

772

65+ (T)

96

191

84

36

80

41

528

Total

274

675

363

197

418

169

2096

Use this table to determine the following probabilities:

  1. (P(C | T))
  1. (P(S | A))
  1. (P(C and T))
  1. (P(R | B))
  1. (P(B | R))
  1. (P(G|S))