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Basic Concepts of Probability
MATH 100, Survey of Mathematical Ideas
J. Robert Buchanan
Department of Mathematics
Fall 2014
J. Robert Buchanan
Basic Concepts of Probability
Background
The early development of the theory of probability was driven
by gambling.
Some occurrences are deterministic (drop a book, it will
hit the floor).
Other occurrences are random (flip a fair coin, the
outcome will be heads half the time).
Probability is the branch of mathematics dedicated to
determining the likelihood of random phenomena.
Any observation of measurement of a random phenomena
is called an experiment. The results of an experiment are
called outcomes. The set of all possible outcomes of an
experiment is called the sample space.
J. Robert Buchanan
Basic Concepts of Probability
Theoretical Probability Formula
If all the outcomes in a sample space S are equally likely, and
E is an event within the sample space, then the theoretical
probability of event E is
P(E) =
number of favorable outcomes
n(E)
=
total number of outcomes
n(S)
J. Robert Buchanan
Basic Concepts of Probability
Example
A group of preschool children includes eight girls and five boys.
One child from the group is randomly selected, with E being the
outcome that the selected child is a girl. What are the
following?
1
The total number of outcomes?
2
The number of favorable outcomes?
3
The number of unfavorable outcomes?
4
The probability of event E?
J. Robert Buchanan
Basic Concepts of Probability
Empirical Probability Formula
Sometimes we do not know that all events in the sample space
occur with equal probability. In this case we may perform an
experiment to approximate the probability of an outcome.
J. Robert Buchanan
Basic Concepts of Probability
Empirical Probability Formula
Sometimes we do not know that all events in the sample space
occur with equal probability. In this case we may perform an
experiment to approximate the probability of an outcome.
If E is an event that may happen when an experiment is
performed, then the empirical probability of event E is
P(E) ≈
number of times event E occurred
.
number of times the experiment was performed
J. Robert Buchanan
Basic Concepts of Probability
Empirical Probability Formula
Sometimes we do not know that all events in the sample space
occur with equal probability. In this case we may perform an
experiment to approximate the probability of an outcome.
If E is an event that may happen when an experiment is
performed, then the empirical probability of event E is
P(E) ≈
number of times event E occurred
.
number of times the experiment was performed
Example
When 200 seeds were planted, 170 of them germinated. Find
the probability that this type of seed will germinate.
J. Robert Buchanan
Basic Concepts of Probability
Law of Large Numbers
As an experiment is repeated more and more times, the
proportion of outcomes favorable to any particular event will
tend to come closer and closer to the theoretical probability of
that event.
J. Robert Buchanan
Basic Concepts of Probability
Example
Suppose three fair coins are tossed.
1
What is the sample space?
2
What is the probability of no heads?
3
What is the probability of exactly one head?
4
What is the probability of exactly two heads?
5
What is the probability of three heads?
J. Robert Buchanan
Basic Concepts of Probability
Example
A family would like to have exactly two daughters. What is the
probability of having exactly two daughters, if the family has
four children all together?
J. Robert Buchanan
Basic Concepts of Probability
Probability in Genetics (1 of 2)
The father of modern genetics (Gregor Mendel) observed the
following inheritance behavior in pea plants.
Pure red plants crossed with pure white plants
produce red plants.
J. Robert Buchanan
Basic Concepts of Probability
Probability in Genetics (1 of 2)
The father of modern genetics (Gregor Mendel) observed the
following inheritance behavior in pea plants.
Pure red plants crossed with pure white plants
produce red plants.
First to Second Generation
Second Parent
r
r
First
R
Rr
Rr
Parent R
Rr
Rr
Each member of the second generation is red, since red is the
dominant gene.
J. Robert Buchanan
Basic Concepts of Probability
Probability in Genetics (2 of 2)
Suppose the plants from the second generation are crossed
with each other.
Second to Third Generation
Second Parent
R
r
First
R
RR
Rr
Parent r
rR
rr
J. Robert Buchanan
Basic Concepts of Probability
Probability in Genetics (2 of 2)
Suppose the plants from the second generation are crossed
with each other.
Second to Third Generation
Second Parent
R
r
First
R
RR
Rr
Parent r
rR
rr
1
What is the probability that a plant in the third generation is
red?
2
What is the probability that a plant in the third generation is
white?
J. Robert Buchanan
Basic Concepts of Probability
Odds
Recall: the probability of an event compares favorable
outcomes to the total number of possible outcomes.
In gambling the number of favorable events is often compared
to the number of unfavorable events.
J. Robert Buchanan
Basic Concepts of Probability
Odds
Recall: the probability of an event compares favorable
outcomes to the total number of possible outcomes.
In gambling the number of favorable events is often compared
to the number of unfavorable events.
Definition
If all outcomes in a sample space are equally likely, with a of
them favorable to event E and the remaining b outcomes
unfavorable to event E, then the odds in favor of E are a to b,
and the odds against E are b to a.
J. Robert Buchanan
Basic Concepts of Probability
Example
A jar contains 5 red balls, 6 yellow balls, and 3 green balls. A
blind-folded person will randomly select one ball from the jar.
1
Find the odds in favor of selecting a red ball.
2
Find the odds against selecting a yellow ball.
3
Find the odds in favor of selecting a green ball.
J. Robert Buchanan
Basic Concepts of Probability
Converting Between Odds and Probabilities
If the odds in favor of event E are a to b, then
P(E) =
J. Robert Buchanan
a
.
a+b
Basic Concepts of Probability
Converting Between Odds and Probabilities
If the odds in favor of event E are a to b, then
P(E) =
a
.
a+b
Example
Find the probability of winning a lottery if the odds in favor of
wining are 1 to 125, 000.
J. Robert Buchanan
Basic Concepts of Probability
Example
Suppose the probability of event E is P(E) = 0.124, what are
the odds in favor of event E?
J. Robert Buchanan
Basic Concepts of Probability
Example
Suppose the probability of event E is P(E) = 0.124, what are
the odds in favor of event E?
P(E) = 0.124 =
31
31
124
=
=
1000
250
31 + 219
The odds in favor of E are 31 to 250 − 31 = 219.
J. Robert Buchanan
Basic Concepts of Probability
Homework
Read Section 12.1.
Exercises: 1, 5, 9, 13, . . . , 45, 49
J. Robert Buchanan
Basic Concepts of Probability