Transcript Probability & Counting Rules

Chapter 4
Created by Laura Ralston
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Classical Probability
Probability Rules
Basic Counting Rules
Additional Counting Techniques
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Probability
◦ What is it?
◦ How is it computed in a variety of important
circumstances?
◦ How is it useful to me?
 From the time you awake until you go to bed, you
make decisions regarding the possible events that are
governed at least in part by chance.
 Should I carry an umbrella today?
 Should I accept that new job?
 Will I have enough gas to get to school?
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In general,
probability is
defined as the
likelihood of an
event occurring
◦ Card Games
◦ Slot Machines
◦ Lotteries

In theory,
probability is the
underlying
foundation on
which inferential
statistics is built
◦ Insurance
◦ Investments
◦ Weather forecasting
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Objectives:
◦ Identify the sample space of a probability event
◦ Calculate basic probabilities

Probability
experiments: any
process in which the
result is random in
nature
◦ Flip a Coin
◦ Roll a die
◦ Answer multiple choice
questions with 4
possibilities
◦ Take a pregnancy test
◦ Predict grade in MATH
2200 for Fall 2008
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Outcome: the
individual result of a
trial in a probability
experiment
Sample Space: the set
of ALL possible
outcomes of a
probability
experiment
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In the previous
examples, the
sample spaces were
found by observation
or reasoning, BUT
what if the
probability
experiment is “more
complex”?
◦ Roll TWO dice
◦ Gender of children if a
family has 3 children
◦ Select card from a
standard 52-card deck
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If the probability
experiment is “more
complex”, we can use
a
◦ Two-way Table
◦ Tree Diagram: device
that organizes the
outcomes of a
probability experiment
with several stages—
tree begins with
outcomes for first
stage and branches for
each additional
possibility

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1) When ordering a pizza with a coupon, you
can have a choice of crusts: thin (T), handtossed (H), or stuffed (S). You can also
choose one topping from the following:
pepperoni (P), ham (M), sausage (G), onion
(O), bell pepper (B), or olives (V)
2) A family has three children. Give the
sample space in regard to the sex of the
children
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Simple Event: an
event with one
outcome
◦ Roll a die and a 6
shows
◦ Flip a coin and a HEAD
shows
◦ Take a pregnancy test
and a NEGATIVE result
shows
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Compound Event: an
event with two or
more outcomes
◦ Roll a die and an odd
number shows (1,3, or
5)
◦ Answer a multiple
choice question with 4
possibilities (a, b, c, or
d)
◦ Predict grade in MATH
2200 (A, B, C, D, or F)
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P: denotes a
probability
A, B, and C:
denotes a specific
event
P(A): is read “the
probability of event
A”
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Classical Probability (aka Theoretical
Probability)
Empirical Probability
Subjective Probability

Uses a probability value based on an
educated guess or estimate, employing
opinions and inexact information
◦ Weather Prediction
◦ Earthquake Prediction
◦ Braves win pennant in 2011 Prediction
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As a probability experiment is repeated again
and again, the relative frequency probability
of an event tends to approach the actual
probability
◦ Conduct experiment: Flip coin 10 times per
individual and record number of times a HEAD
occurs
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First type of probability studied in 17th-18th
centuries
Most precise type of probability
Assumes that all outcomes in the sample
space are equally likely to occur
Final results can be expressed as fractions,
decimals, or percentages
◦ Always simply fractions
◦ Round decimals to three places. If extremely small,
it is okay to round the decimal to the first nonzero
digit.
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3) If a die is rolled one time, find the
probability:
◦ a. P(getting a 4)
◦ b. P(getting an even number)
◦ c. P(getting a number greater than 3 or an odd
number)

4) A couple has three children. Find the
probability:
◦ a. P(all boys)
◦ b. P(exactly two boys or two girls)
◦ c. P(at least one child of each gender)

In a college class of 250 graduating seniors,
50 have jobs waiting, 10 are going to medical
school, 20 are going to law school, and 80
are going to various other kinds of graduate
schools.
◦ 5) How many have no jobs or are not attending
graduate school?
◦ 6) Select one graduate at random. Find the
probability:
 a. P(student is going to graduate school)
 b. P(student will have to start paying back loans after
6 months---does not continue in school)
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