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8.4.1: Conditional Probability (Exercises)

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    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. \(P(G|S)\)

    This page titled 8.4.1: Conditional Probability (Exercises) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Rupinder Sekhon and Roberta Bloom via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.