Download Activity 5 - Saint Mary`s College

Math 104
ACTIVITY 5: Probabilities based on “equally likely outcome” models
Why
We now have enough of the basic counting rules (there are many more rules, and more complicated rules) to
make use of them in calculating probabilities in the equiprobable case (all outcomes equally likely). This will
require understanding the description of the experiment and the desired event clearly enough to count all the
possible outcomes (the size of the sample space) and also count the number of desirable outcomes (the event
we are looking for).
LEARNING OBJECTIVES
1. Be able to work effectively as a team, using the team roles
2. Be able to identify and count the outcomes for an experiment from a description of the experiment.
3. Be able to identify and count the outcomes that produce a particular event, from a description of the experiment
and the event.
4. Be able to recognize and use the equiprobable outcomes model to calculate probabilities.
CITERIA
1. Success in working as a team and in fulfilling the team roles.
2. Success in involving all members of the team in the conversation.
3. Success in distinguishing and using the appropriate counting rules.
4. Success in calculating the probabilities requested.
RESOURCES
1. The course syllabus
2. The team role desk markers (handed out in class for use during the semester)
3. The Information booklet for Math 104 Particularly pages 3 (Counting Techniques) and 4 (Strategy for analyzing a
Counting Problem)
4. Your text—Sections 2.1 - 2.4
5. 50 minutes
PLAN
1. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3.
2. Work through the exercises given here - be sure everyone understands all results & procedures.
3. Assess the team’s work and roles performances and prepare the Reflector’s and Recorder’s reports including team
grade.
DISCUSSION
For counting problems, the outline in the Information Booklet is your best summary of our techniques, except
that it does not directly mention counting by using the complement — which should always be considered
when there are cases.
With equally likely outcomes, remember that the probability of any event E is given by
number of outcomes that fit E
P r(E) =
. Calculation requires solving two counting problems
total number of outcomes for the experiment
which are related, but not the same. For the denominator (total number of outcomes) we must ignore the
result we want and count all the possibilities, and for the denominator we must count all (and only) those
outcomes that give the result we want. Situations that fit the “equally likely outcomes” condition often come
up we select people or groups at random. (“At random” is being used as a technical term meaning that any
group is as likely as any other to be selected.)
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EXERCISE
1. A study committee of 6 people is to be formed from a work group consisting of 8 women and 12 men. Suppose the
committee members are to be assigned at random.
(a) What does an outcome of this experiment look like? (What are we counting when we count the sample space)?
(b) How many different committees are possible?
(c) How many of these possible committees include at least one woman (Complement counting is probably easiest,
here)?
(d) What is the probability the committee will include at least one woman?
(e) What is the probability the committee will include more women than men (This involves more than one case)?
2. A radio station manager must name radio show hosts for four time slots: 5 pm, 7 pm, 9 pm and 11 pm. There are
8 men and 7 women who have applied for the slots—all are equally qualified and the manager will assign people to
the slots at random.
(a) What does an outcome of this experiment look like? (What will the manager have, at the end? What are we
counting when we count the sample space?)
(b) How many different assignments are possible (it matters which slot a person assigned to)?
(c) How many different assignments are there in which a man is assigned to the 5 pm time slot?
(d) What is the probability that a man is assigned to the 5 pm time slot?
(e) What is the probability that the schedule will have a man at 5 and all other slots filled by women?
3. There are 8 sophomores, 7 juniors and 5 seniors on the College of Knowledge Quiz Bowl team. Four team members
are to be selected to participate in a tournament.
(a) How many selections of four are possible?
(b) How may selections are there in which the people chosen are all from the same class (all soph or all jr or all
sr)?
(c) If the selection is random, what is the probability the students will all be from the same class?
(d) What is the probability the students will all be sophomores or seniors (no juniors)?
4. Five cards are to be drawn from a well-shuffled (another way to say all selections equally likely) standard deck of
cards.
(a) What is the probability of getting all red cards?
(b) What is the probability of getting 3 spades, one heart, one diamond?
(c) What is the probability of getting exactly 3 Aces?
READING ASSIGNMENT (in preparation for next class meeting)
Re-read Section 2.4 in the text
SKILL EXERCISES:(hand in - individually - at next class meeting)
p.82 # 4-14
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