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Introduction
„
Probability
„
Introduction;
Experiments, Outcomes, & Events
Probability of an Event
The probability of an event is a number
between 0 and 1 that expresses the longrun likelihood that the event will occur.
„ An event having probability .1 is rather
unlikely to occur.
„ An event with probability .9 is very likely
to occur.
„ An event with probability .5 is just as likely
to occur as not.
„
Example: Experiment, Trial and Outcome
„
Experiment 1: Flip a coin
Trial: One coin flip
Outcome: Heads
„
Experiment 2: Allow a conditioned rat to run a
maze containing three possible paths
Trial: One run
Outcome: Path 1
„
Experiment 3: Tabulate the amount of rainfall
in New York, NY in a year
Trial: One year
Outcome: 37.23 in
In this chapter, we discuss probability, which is
the mathematics of chance.
Many events in the world around us exhibit a
random character, but by repeated observations
of such events we can often determine longterm patterns (despite random, short-term
fluctuations). Probability is the branch of
mathematics devoted to the study of such
events.
Experiment, Trial, & Outcome
An experiment is an activity with an
observable outcome.
„ Each repetition of the experiment is called
a trial.
„ In each trial we observe the outcome of
the experiment.
„
Sample Space
The set of all possible outcomes of an
experiment is called the sample space of the
experiment. So each outcome is an element of
the sample space.
There are two types of sample spaces:
finite and infinite.
Note: The sample space of an event is
equivalent to the universal set.
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Example: Sample Space
An experiment consists of throwing two
dice, one red and one green, and observing
the numbers on the uppermost face on
each. What is the sample space S of this
experiment?
„ Each outcome of the experiment can be
regarded as an ordered pair of numbers,
the first representing the number on the
red die and the second the number on the
green die.
Example: Sample Space
„
Event
„
S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6) (3,1),
(3,2), (3,3), (3,4), (3,5), (3,6) (4,1), (4,2),
(4,3), (4,4), (4,5), (4,6) (5,1), (5,2), (5,3),
(5,4), (5,5), (5,6) (6,1), (6,2), (6,3), (6,4),
(6,5), (6,6)}
Example: Event
For the experiment of rolling two dice,
describe the events
„ E1 = {The sum of the numbers is greater
than 9}
„
An event E is a subset of the sample
space.
„ We say that the event occurs when the
outcome of the experiment is an element
of E.
„
„
E2 = {The sum of the numbers is 7 or 11}.
Events as Sets
Special Events
Let E and F be two events of the sample space S.
„
„
Let S be the sample space of an experiment.
The event corresponding to the empty set, Ø, is
called the impossible event, since it can never
occur.
The event corresponding to the sample space
itself, S, is called the certain event because the
outcome must be in S.
The event where either E or F or both
occurs is designated by E ∪ F.
„ The event where both E and F occurs is
designated by E ∩ F.
„ The event where E does not occur is
designated by E '.
„
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Example: Events As Sets
For the experiment of rolling two dice, let
E1 = “The sum of the numbers is greater than 9”
and
E3 = “The numbers on the two dice are equal”.
Determine the sets:
„ E1 ∪ E3
Mutually Exclusive Events
„
„
„
„
Let E and F be events in a sample space S. Then
E and F are mutually exclusive (or disjoint) if
E∩F=Ø
If E and F are mutually exclusive, then E and F
cannot simultaneously occur; if E occurs, then F
does not; and if F occurs, then E does not.
E1 ∩ E3
Example
Example: Mutually Exclusive Events
„
„
„
For the experiment of rolling two dice, which of
the following events are mutually exclusive?
E1 = “The sum of the dots is greater than 9”
E2 = “The sum of the dots is 7 or 11”
E3 = “The dots on the two dice are equal”
„
A letter is selected at random from the word
“ALABAMA.”
a.) What is the sample space for this
experiment?
b.) Describe the event “the letter chosen is a
vowel” as a subset of the sample space.
Example
„
Example
An experiment consists of tossing a coin four times
and observing the sequence of heads and tails.
„
a.) What is the sample space of this experiment?
b.) Determine the event E 1 = “more heads
than tails occur.”
c.) Determine the event E 2 =“the first toss is a tail.”
Suppose that you observe the time (in
minutes) that it takes a bank teller to deal
with a customer.
Describe the sample space.
Is this a finite or infinite sample space?
d.) Determine the event E 1 ∩ E 2.
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Example: The Game of Clue
Example
„
Consider the following events:
„
A = “a person has a dog”
B = “a person is taking a math class”
C = “ a person does not have any pets”
D = “a person is taking an English class”
„
Which, if any, of the events would be mutually
exclusive? Explain.
Anthony E. Pratt, then inventor of the game
Clue, died in 1996. Clue is a board game in
which players are given the opportunity to
solve a murder that has six suspects, six
possible weapons, and nine possible rooms,
where the murder may have occurred.
The six suspects are Colonel Mustard, Miss
Scarlet, Professor Plum, Mrs. White, Mr.
Green, and Mrs. Peacock.
Example: The Game of Clue (continued)
„
„
„
How could a sample space be formed with the
entire solution to the murder, giving murderer,
weapon, and site?
How many outcomes would the sample space
have?
Let E be the event the murder occurred in the
library. Let F be the event that the weapon was
a gun. Describe E ∪ F and E ∩ F.
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