ABILENE CHRISTIAN UNIVERSITY
Department of Mathematics
Probability and Chance
Section 14.1
Dr. John Ehrke
Department of Mathematics
Fall 2012
ABILENE CHRISTIAN UNIVERSITY
DEPARTMENT OF MATHEMATICS
Definitions
Definition (Random Experiment)
An experiment that does not yield the same results even if repeated under
same conditions. (i.e flipping a coin, rolling dice, etc...)
Definition (Sample Space)
The set of all possible outcomes (events) of an experiment in which each
trial produces one outcome. A sample space is called an equally likely space
if every outcome has an equal chance of occurring.
Definition (Event)
Any subset of the sample space including the empty set, ∅, and the sample
space S are called events. A simple event contains only one outcome in its
set, while compound events can contain multiple elements from the sample
space.
Slide 2/10 — Dr. John Ehrke — Lecture 3 — Fall 2012
ABILENE CHRISTIAN UNIVERSITY
DEPARTMENT OF MATHEMATICS
Simple Events and Probability
Definition (Probability of an Event)
Denoted, P(E), the probability of the event E is the sum of the probabilities
of the outcomes (simple events) that make up E. When the probabilities of
the simple events are known the probability of the event E is given by
P(E) =
number of ways E occurs
.
number of outcomes in the sample space
Example
Suppose your random experiment is rolling a pair of six-sided dice and
recording the results.
• What are the simple events?
• Is this an equally likely space?
• What is the probability of rolling a sum of six?
• How does this change if we record the sum of the two dice instead?
Slide 3/10 — Dr. John Ehrke — Lecture 3 — Fall 2012
ABILENE CHRISTIAN UNIVERSITY
DEPARTMENT OF MATHEMATICS
Empirical v Theoretical Probability
Example
Suppose we are performing an experiment by flipping a fair coin and recording
whether the coin came up heads or tails. Find a formula for the relative and absolute
frequency of tossing a head (h) after (n) trials.
Solution: If you were to flip a coin 10,000 times you would expect the number of
heads to be approximately equal to the number of tails when using a fair count.
Therefore the absolute difference between heads and tails can be expressed as
Absolute Frequency Difference = h −
n
.
2
For example if we perform, n = 1000 trials and observe h = 500 heads the absolute
difference would be 500 − (1000/2) = 0. In comparison, the relative frequency
difference can be computed as
Relative Frequency Difference =
We should expect the difference converges to zero.
Slide 4/10 — Dr. John Ehrke — Lecture 3 — Fall 2012
h
1
− .
n
2
ABILENE CHRISTIAN UNIVERSITY
DEPARTMENT OF MATHEMATICS
Polling Question # 7
Assume that the probability of having a boy is the same as having a girl.
Under this assumption, in a three child family, what is the probability that
at least one of the children is a girl?
(a) 7/8
(b) 1/8
(c) 1/2
(d) 5/8
Slide 5/10 — Dr. John Ehrke — Lecture 3 — Fall 2012
ABILENE CHRISTIAN UNIVERSITY
DEPARTMENT OF MATHEMATICS
Polling Question #8
Suppose you roll two fair six-sided dice. What is the probability of rolling
no twos?
(a) 11/36
(b) 25/36
(c) 2/36
(d) 8/36
Slide 6/10 — Dr. John Ehrke — Lecture 3 — Fall 2012
ABILENE CHRISTIAN UNIVERSITY
DEPARTMENT OF MATHEMATICS
Polling Question #9
A shipment of 20 light bulbs arrives at a warehouse for inventory. From
past experience 3 of the bulbs will be defective. What is the probability if 5
bulbs are chosen at random for testing that none of the five will be
defective? (Questions like this are very important in quality assurance
testing for businesses.)
(a)
C(20, 3)
C(20, 5)
(b)
P(20, 3)
P(20, 5)
(c)
C(15, 5)
C(20, 5)
(d)
C(17, 5)
C(20, 5)
Slide 7/10 — Dr. John Ehrke — Lecture 3 — Fall 2012
ABILENE CHRISTIAN UNIVERSITY
DEPARTMENT OF MATHEMATICS
Odds
Expressions of likelihood are often given as odds, such as 50:1. The
definitions we will use are given below:
Definition (Odds Against)
The odds against an event A occurring are given by n(A) : n(A) usually
expressed in the form a : b where a = n(A) is the number of outcomes not
resulting in A.
Definition (Odds in Favor)
The odds in favor of the event A occurring is simply the opposite of odds
against. That is, if the odds against an event A are a : b, then the odds in
favor of the event A are b : a.
If the actual odds against the event A are a : b, then
P(A) =
Slide 8/10 — Dr. John Ehrke — Lecture 3 — Fall 2012
b
a+b
or
P(A) =
a
.
a+b
ABILENE CHRISTIAN UNIVERSITY
DEPARTMENT OF MATHEMATICS
Class Discussion Question #1
Suppose you go to the races to place a $100 bet on a horse named Hairy
Plotter who is running at 5:2 odds against.
(a) What is the probability the horse will win the race?
(b) In the event the horse wins the race, what is your expected payoff?
(c) Suppose another horse was odds on favorite at 4:5 against, what would
the expected payoff for a $250 bet be in this case?
Slide 9/10 — Dr. John Ehrke — Lecture 3 — Fall 2012
ABILENE CHRISTIAN UNIVERSITY
DEPARTMENT OF MATHEMATICS
Class Discussion Question #2
A boy and a girl are playing a game in which both simultaneously call out a
number from 1 through 3. Find the probability for each of the following:
1
Both call out an odd number.
2
Both call out the same number.
3
One of them calls 2, while the other does not.
Slide 10/10 — Dr. John Ehrke — Lecture 3 — Fall 2012