Download TUTORIAL 4 - Probability Distributions

Slide 1
Statistics Workshop
Tutorial 4
Probability
• Probability Distributions
•
Slide 2
Probability
Created by Tom Wegleitner, Centreville, Virginia
Definitions
Slide 3
 Event
Any collection of results or outcomes of a
procedure.
 Simple Event
An outcome or an event that cannot be further
broken down into simpler components.
 Sample Space
Consists of all possible simple events. That is,
the sample space consists of all outcomes that
cannot be broken down any further.
Copyright © 2004 Pearson Education, Inc.
Notation for
Probabilities
Slide 4
P - denotes a probability.
A, B, and C - denote specific events.
P (A) -
denotes the probability of
event A occurring.
Copyright © 2004 Pearson Education, Inc.
Basic Rules for
Computing Probability
Slide 5
Rule 1: Relative Frequency Approximation
of Probability
Conduct (or observe) a procedure a large
number of times, and count the number of
times event A actually occurs. Based on
these actual results, P(A) is estimated as
follows:
P(A) =
number of times A occurred
number of times trial was repeated
Copyright © 2004 Pearson Education, Inc.
Basic Rules for
Computing Probability
Slide 6
Rule 2: Classical Approach to Probability
(Requires Equally Likely Outcomes)
Assume that a given procedure has n different
simple events and that each of those simple
events has an equal chance of occurring. If
event A can occur in s of these n ways, then
s
=
P(A) =
n
number of ways A can occur
number of different
simple events
Copyright © 2004 Pearson Education, Inc.
Law of
Large Numbers
Slide 7
As a procedure is repeated again and
again, the relative frequency probability
(from rule 1) of an event tends to
approach the actual probability.
Copyright © 2004 Pearson Education, Inc.
• A simple example of randomness involves a coin
toss. The outcome of the toss is uncertain. Since
the coin tossing experiment is unpredictable, the
outcome is said to exhibit randomness.
• Even though individual flips of a coin are
unpredictable, if we flip the coin a large number of
times, a pattern will emerge. Roughly half of the
flips will be heads and half will be tails.
Illustration of Law of Large Numbers
Slide 9
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Probability Limits
Slide 10
 The probability of an impossible event is 0.
 The probability of an event that is certain to
occur is 1.
 0  P(A)  1 for any event A.
Copyright © 2004 Pearson Education, Inc.
Possible Values for
Probabilities
Figure 3-2
Copyright © 2004 Pearson Education, Inc.
Slide 11
Slide 13
Probability Distributions
Created by Tom Wegleitner, Centreville, Virginia
Copyright © 2004 Pearson Education, Inc.
Overview
Slide 14
This chapter will deal with the construction of
probability distributions
by combining the methods of descriptive
statistics presented in Chapter 2 and those
of probability presented in Chapter 3.
Probability Distributions will describe what
will probably happen instead of what
actually did happen.
Copyright © 2004 Pearson Education, Inc.
Combining Descriptive Methods
and Probabilities
Slide 15
In this chapter we will construct probability distributions
by presenting possible outcomes along with the relative
frequencies we expect.
Figure 4-1
Copyright © 2004 Pearson Education, Inc.
Definitions
Slide 16
 A random variable is a variable (typically
represented by x) that has a single
numerical value, determined by chance,
for each outcome of a procedure.
 A probability distribution is a graph,
table, or formula that gives the
probability for each value of the
random variable.
Copyright © 2004 Pearson Education, Inc.
Definitions
Slide 17
 A discrete random variable has either a finite
number of values or countable number of
values, where “countable” refers to the fact
that there might be infinitely many values, but
they result from a counting process.
 A continuous random variable has infinitely
many values, and those values can be
associated with measurements on a
continuous scale in such a way that there are
no gaps or interruptions.
Copyright © 2004 Pearson Education, Inc.
Graphs
Slide 18
The probability histogram is very similar to a relative frequency
histogram, but the vertical scale shows probabilities.
Figure 4-3
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Requirements for
Probability Distribution
Slide 19
 P(x) = 1
where x assumes all possible values
0  P(x)  1
for every individual value of x
Copyright © 2004 Pearson Education, Inc.
Mean, Variance and
Standard Deviation of a
Probability Distribution
Slide 20
µ =  [x • P(x)]
Mean
 =  [(x – µ) • P(x)]
Variance
 =  [x • P(x)] – µ
Variance (shortcut)
2
2
2
2
2
 =  [x 2 • P(x)] – µ 2
Standard Deviation
Copyright © 2004 Pearson Education, Inc.
Identifying Unusual Results
Range Rule of Thumb
Slide 21
According to the range rule of thumb, most
values should lie within 2 standard deviations
of the mean.
We can therefore identify “unusual” values by
determining if they lie outside these limits:
Maximum usual value = μ + 2σ
Minimum usual value = μ – 2σ
Copyright © 2004 Pearson Education, Inc.
Identifying Unusual Results
With Probabilities
Slide 22
Rare Event Rule
If, under a given assumption (such as the
assumption that boys and girls are equally likely),
the probability of a particular observed event (such
as 13 girls in 14 births) is extremely small, we
conclude that the assumption is probably not
correct.
 Unusually high: x successes among n trials is an
unusually high number of successes if P(x or
more) is very small (such as 0.05 or less).
 Unusually low: x successes among n trials is an
unusually low number of successes if P(x or
fewer) is very small (such as 0.05 or less).
Copyright © 2004 Pearson Education, Inc.
Slide 23
Now we are ready for
Part 14 of Day 1