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Section 91 – How Probabilities are Determined
Experiment:
An activity whose results can be observed and recorded.
Each possible result of an experiment is called an
outcome.
Sample Space:
The set of all possible outcomes for an experiment.
Example
:
What is the sample space for flipping a coin one time?
What is the sample space for rolling a die?
Event:
Any subset of a sample space is an event
Ex
:
Rolling an odd # {1, 3, 5}
Example
:
Suppose we toss a fair coin 3 times and record the results.
a)
Find the sample space for this experiment
b)
Find the event A of tossing no heads
c)
Find the event B of tossing a tail on the first toss
1
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View Full Document Determining Probabilities
Experimentally:
When a probability is determined by observing outcomes of experiments
P(A)
denotes the “probability of event A occurring”
Law of Large Numbers (Bernoulli’s Theorem)
:
If an experiment is repeated a large number of
times, the experimental probability of a particular outcome approaches a fixed number as the
number of repetitions increases.
The fixed number mentioned above is “theoretical probability”
We assign
theoretical probabilities
to the outcomes under ideal conditions.
When one outcome is just as likely as another (as in coin tossing or rolling a die), the outcomes are
called
equally likely
.
Probability Rules
:
An impossible event has probability ________.
An event certain to occur has probability ________.
For any event A,
____ ≤ P(A) ≤ ____ .
Example:
Let set
A = {1, 2, 3, 4, 5, 6, 7, 8}.
Calculate the
probability
for each of the following events:
a)
A = an even number is drawn
b)
B = a number more than 5 is drawn
c)
C = a prime number is drawn
d)
D = an even number that is a multiple of three is drawn
e)
E = drawing either a prime or a composite number
2
Example
:
In a movie theater, 50 people have popcorn and 35 people do not.
What is the
probability that a randomly selected person has popcorn?
Example
:
A multiple choice test has 5 choices for each problem.
What is the probability that if you
guess, you will guess incorrectly?
Example
:
Roll 2 dice and record their sums in the following table.
a)
What is the probability of getting a sum of 9?
b)
What is the probability of getting a sum of 7?
+
1
2
3
4
5
6
1
2
3
4
5
6
3
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INTERSECTION
:
P(A
and
B) =
(
)
P A
B
∩
UNION
:
P(A
or
B) =
(
)
P A
B
∪
P(A and B) = P(A and B both occur)
P(A or B)
=
P(A occurs or B occurs or both)
When finding the probability that event A occurs OR
event B occurs, find the total number of ways
that A can occur and the number of ways B can occur, but find the total in such a way that no
outcome is counted more than once
!!
The Addition Rule
:
(
)
( )
( )
(
)
∪
=
+

∩
P A
B
P A
P B
P A
B
Example
:
There are 30 students in a shop class.
18 build bird houses,
8 build mailboxes,
3 build
both.
Set up a Venn Diagram to illustrate.
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This note was uploaded on 01/06/2012 for the course MATH 1351 taught by Professor Lisajuliano during the Spring '12 term at Collins.
 Spring '12
 LisaJuliano
 Geometric Probability

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