Unit Plan

Compound Probabilities

Objectives

In this unit, multistage experiments introduce students to dependence, mutual exclusion, expected value, and the relation of those concepts to addition and multiplication principles. This unit contains activities that use playing cards and number cubes. Students will:
  • learn how to use compound probability and how to distinguish it from simple probability.

  • understand the differences between independent and dependent events and how to use them to determine probability.

  • learn about the idea of replacement and nonreplacement (whether an object is put back for independent events or kept out for dependent events).

  • use the Fundamental Counting Principle to determine possible outcomes.

  • find the odds of an event and the expected value.

  • Prerequisite Knowledge:

    • Union, Intersection, and Appropriate Notation

Essential Questions

  • What differentiates an independent event from a dependent event and how are the probabilities of each calculated?

Related Unit and Lesson Plans

Related Materials & Resources

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Formative Assessment

  • View

    Multiple Choice Items:

    Use the following information to answer questions 1–3.

    Isaac works in an auto parts store and he keeps track of the inventory. He works hard but doesn’t always save his files on the right server. The probability he finds File A on the first try is 0.40. The probability he finds File B on the first try is 0.20. P(C) = 0.20; P(D) = 0.15; and the P(F) = 0.05.

    1. What is the probability that Isaac finds file C on the first try and File A on the first try?

    A

    .60

    B

    .08

    C

    .2

    D

    2

    2. What is the probability that Isaac finds File D on the first try or finds File F on the first try?

    A

    .20

    B

    .0075

    C

    .1925

    D

    .10

    3. What is the probability that Isaac finds File A on the first try, given he found File D on the first try?

    A

    .40

    B

    .06

    C

    .15

    D

    .55


    Use the following information to answer questions 4–6.

    A family has three children.

    4. How many possible outcomes are there for the gender of three children?

    A

    4

    B

    2

    C

    6

    D

    8

    5. What is the probability that of the three children, at least two of them are boys?

    A

    1/2

    B

    1/8

    C

    1/4

    D

    3/8

    6. What is the probability the family’s third child is a girl, given their first child was a boy?

    A

    1/4

    B

    1/2

    C

    1/8

    D

    3/8

    7. If the probability of winning a random drawing is , what are the odds against winning?

    A

    1:2000

    B

    2000:1

    C

    1:1999

    D

    1999:1

    8. In a standard 52-card deck, the odds of drawing a face card are 3:10; what is the probability of drawing a face-card?

    A
    3/10

    B

    3/52

    C

    3/13

    D

    3/26

    9. When Jason flips a fair coin, there is an equal probability of heads-up or tails-up. Jason flipped the coin 3 times. It landed heads-up all 3 times. What is the probability the coin will land heads-up the next time Jason flips the coin?

    A

    1/2

    B

    1/4

    C

    1/8

    D

    1/16

    Multiple choice key:

    1. B

    2. A

    3. A

    4. D

    5. A

    6. B

    7. D

    8. C

    9. A


    Short Answer Items:

    10. Make a tree diagram/list of all possible gender outcomes of two sets of twins.

    11. Suppose that in a certain high school class, 50% of the students have brown eyes, 30% have blue eyes, 12% have green eyes, and 8% have hazel eyes. What is the probability that two students chosen at random have the same eye color?

    12. Consider the nine multiple-choice questions at the beginning of the test.

    a) What is the probability of answering one correctly by randomly guessing?

    b) What is the probability of answering them all correctly by guessing?

    13. The table shows some population percentages for the United States based on the year 2000 census.

    U.S. Population Percentages according to age in 2000

    Gender

    Under 18

    18 and older

    Male

    26.8

    73.2

    Female

    24.6

    75.4


    1. A random U.S. adult (age 18 and older) is selected; find the probability that the adult is male.

    b) A random U.S. female is selected; find the probability that she is under 18.



    Short-answer key and Scoring Rubric:

    10. Answer: Tree/sample space should show the following outcomes: (16)

    MMMM, FMMM, MFMM, MMFM, MMMF, FFFF, MFFF, FMFF, FFMF, FFFM, MMFF, FFMM, FMFM, MFFM, FMMF, MFMF

    11. Answer: (.5 × .5) + (.3 × .3) + (.12 × .12) + (.08 × .08) = .3608

    12. Answer: a) 1/4   b) (1/4) 9 = .000003815

    13. Answer: a) .50 × .732 = .366 or 36.6%

    b) .50 × .246 = .123 or 12.3%


    Points

    Description

    2

    • Response is complete, correct, and detailed.

    • Student demonstrates thorough understanding of compound probabilities.

    • Supporting work is complete and correct.

    1

    • Response is partially correct or true but does not answer the specific question or is correct but lacking detail.

    • Student demonstrates partial understanding of compound probabilities.

    • Some supporting work is provided.

    0

    • Response is incorrect.

    • Student demonstrates no understanding of compound probabilities.

    • Supporting work not provided, unclear, or incorrect.











    Performance Assessment:

    The game 500 is played with six standard six-sided number cubes and two or more players. The basic rules for this game can be found at http://en.wikipedia.org/wiki/Farkle. 500 is a variant of Farkle in which each player must continue rolling the remaining number cubes on his/her turn until reaching an initial threshold of at least 500 points. After the threshold is reached, the player may stop at any time and may play freely (with no further threshold) on subsequent turns. Players who “farkle” on their first turn, prior to reaching the threshold, must continue to strive for that threshold on subsequent turns. 500 is played to a winning score of 500 points.

    Farkle is played by two or more players, with each player in succession having a turn at throwing the number cubes. Each player’s turn results in a score, and the scores for each player accumulate to some winning total.

    At the beginning of each turn, the player throws all six 6-sided number cubes.

    After each throw, one or more scoring number cubes must be set aside. 1s and 5s are always scorable; 2s, 3s, 4s, and 6s are only scorable when they occur in a set of three or more
    (e.g., 2, 2, 2).

    The player may then either end his/her turn and bank the score accumulated so far, or continue to throw the remaining number cubes. If the player has scored all six number cubes, s/he has “hot cubes” and may continue his/her turn with a new throw of all six number cubes, adding to the score already accumulated.

    If no number cubes score in any given throw, the player has “farkled” and all points for that turn are lost.

    At the end of the player’s turn, the number cubes are handed to the next player in succession (usually in clockwise rotation), and that person takes a turn. Once a player has achieved a winning point total, every other player has one last turn to score enough points to surpass that high score.

    Here is an example of play:

    Player 1

    First roll: 1, 2, 4, 4, 6, 6               Sets aside: 1 (100 points)

    Second roll: 3, 5, 5, 6, 6               Sets aside: 5 (50 points)

    Third roll: 1, 1, 2, 5                     Sets aside: 1, 1 (200 points)

    Fourth roll: 3, 4 (farkle)

    Since only 350 total points have been accumulated, Player 1 has not met the initial 500-point threshold, and scores 0. Player 1 must attempt to reach this threshold again on his/her next turn.

    Player 2

    First roll: 2, 3, 3, 3, 5, 6 Sets aside: 3, 3, 3 (300 points)

    Second roll: 1, 4, 5 Sets aside: 1, 5 (150 points)

    Third roll: 1 Sets aside: 1 (100 points)

    Since 550 total points have been accumulated, Player 2 now has the option to stop (recording a score of 550 for the first turn), or reroll all six “hot cubes” and attempt to add to the first turn’s score. If, however, Player 2 should farkle after rerolling, all scoring would be lost and the threshold would be, in effect, on Player 2’s next turn.

    Performance Task:

    Determine whether there is a winning strategy when playing 500.

    Your analysis of the game should include:

    • experimental data collected from playing 500 using various strategies;

    • varying analysis for different game situations, such as aiming for the 500-point threshold, how many scoring number cubes to set aside, whether to set aside or break up triples, how to play when nearing 5,000 points, etc.;

    • identifying various types of events that could happen in the game, classifying each as dependent or independent; and

    • the calculation of theoretical probabilities, using the concepts learned in the Compound Probability unit.

    (Recommendation to the teacher: On the day the Performance Task is due, set up a 500 tournament in which students are forced to use their winning strategy!)

    Score

    Experimental Data

    Analysis

    Event Identification

    Theoretical Calculation


    20%

    50%

    10%

    20%

    4

    Must indicate that a sufficient number of turns have been taken to get passed rolling a losing number. More than sufficient data has been collected in order to observe patterns and draw conclusions.

    Four or more strategies are exhaustively tested.

    Analysis covers almost every conceivable game situation, even beyond those given as examples.

    The probability-based justifications are without error.

    The analysis is good enough to be part of a published strategy guide for the game.

    List of events is exhaustive.

    Sequences of events are properly identified as being dependent or independent.

    Theoretical probability calculations are correct and relevant.

    The entire base of learning from the unit is displayed, including advanced topics like expected value and combinations.

    3

    Sufficient data has been collected in order to observe patterns and draw conclusions.

    At least three different strategies are part of the experiment.

    Analysis covers at least the game situations mentioned in the examples in the Performance Task description.

    It uses the experimental data and theoretical probabilities as strong supporting justification.

    The existence/non-existence of winning strategies is clear.

    A thorough list of related simple events has been created.

    Sequences of events are properly identified as being dependent or independent.

    Theoretical probability calculations are correct and relevant.

    They demonstrate the addition and multiplication rules of probability, using tools like probability trees and definition-based formulas.

    Performance Assessment Scoring Rubric:

    2

    The experiment includes only two strategies, or sample size is insufficient to draw conclusions for at least one of the strategies.

    At least one of the example game situations is not analyzed.

    Analysis may be a bit shaky, with incomplete justification, but the conclusion about the existence of winning strategies is still clear.

    List of events is missing a few key events.

    One or two events may be misidentified.

    Theoretical probability calculations may show small errors.

    Major topics from the unit are identifiable in the student’s work.

    1

    Conclusions are weak and unsubstantiated, or only one strategy is tested.

    Multiple example-game situations are not analyzed; analysis may simply be general in nature.

    Analysis is shoddy, trivial, or largely incorrect.

    Conclusion may be unclear.

    Many obvious events are missing from the list, or misidentification of events is extensive and makes it clear that the student does not understand event dependence.

    Theoretical probability calculations are largely incorrect or irrelevant to the Performance Task.

    Major topics from the unit may be missing entirely.

    0

    Experimental data is not included.

    Performance Task is never completed.

    List of events is not included.

    Theoretical probabilities are not present or are entirely irrelevant to the Performance Task.

DRAFT 05/28/2010
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