Unformatted text preview: 2 Random Experiments Sample Spaces • D fi i i
Definition: • D fi i i
Definition: 3 • E.g. if we are to flip a (typical) coin once, the
possible outcomes are “heads” or “tails” (that we
heads
tails
can denote H and T respectively) to give:
S={H,T}
• If our experiment involves flipping a single coin
twice
t i we get:
t
S = { HH, HT, TH, TT } 4 More “Sample Space” Examples More “Sample Space” Examples
• Example 21 (continued) Example 21 5 Discrete v. Continuous Sample
v
Spaces 6 More “Sample Space” Examples
Example 22 • Discrete:
– Th number of members of th sample space is fi it or
The
b
f
b
f the
l
i finite
countably infinite • C ti
Continuous:
– The number of members of the sample space is infinite
and uncountable
d
t bl
• Notes:
– A “countably infinite” set has the same number of members as the set of
positive integers (i.e. you can count them but it’ll take you forever)
– An uncountable set has the same number of members as a portion of the
real line (e.g. there’s an uncountable number of real numbers between 0
and 1) 7 8 The Outcomes in a Sample Space
Are Not Necessarily Equally Likely
y q
y
y More “Sample Space” Examples • In general, the outcomes in a sample space are not equally likely: Example 22 (extended) – If the coin used in our coinflipping examples is “fair”, each outcome is
pp g
p
f
equally likely
– In all of our other examples so far, the outcomes are not equally likely • Another example with unequally likely outcomes:
– Suppose we have a population of 100 items, n of which are defective, and
we draw a sample of two items
– D
Denoting a “good” item by “g” and a defective item by “d”, the sample
i
“ d” i
b “ ” d d f i i
b “d” h
l
space is as follows (provided that n>=2): •
• Where “n” means a failed connector and “y” an
acceptable one
• In this case, the sample space is discrete ( the
,
p p
(as
number of outcomes is countably infinite) S = {gg, gd, dg, dd} ... if the ordering of the samples is important
or
S = {gg, gd, dd} ... if the ordering of the samples is unimportant 9 Sampling With (and Without)
Replacement Q. what’s the sample space if there is only a single defective item in the
population? 10 Visualizing a Sample Space via a
Tree Digram • In experiments where samples are taken from a population,
it is significant whether or not this is done with or without
replacement: • When a sample space can be constructed in several steps
(or stages) we can represent it using a tree diagram
stages),
• Construction procedure for a tree diagram: – When sampling with replacement, the sampled item is replaced
into the population before the next sample is drawn
– When sampling without replacement, sampled items are not
returned to the population
t
d t th
l ti – Start at the root of the tree (though this is typically at the top of the
diagram, not the bottom)
– Draw a branch (from the root) to represent each of the n1 outcomes
at the first stage
– From the end of each branch, draw a branch to represent each of
the n2 outcomes at the second stage
– Continue in a similar fashion for all subsequence stages • E.g. if two items are sampled (
g
p (one at a time) from the
)
population {a, b, c} the sample space would be one of:
– Swithout = {ab, ac, ba, bc, ca, cb}
– Swith = {aa, ab, ac, bb, ba, bc, cc, ca, cb}
11 12 A Tree Diagram Example A Tree Diagram Example Example 23 Figure 25 Tree diagram for three messages.
13 Events 14 Event (Set) Operators • D fi i i
Definition:
•
• An event typically represents a collection of
related outcomes that may be of signifcance
• Since an event is a set (remember the sample
(remember,
space is a set), we can manipulate events using set
operators to form other events of significance
15 16 An Event Example Mutually Exclusive Events
• D fi i i
Definition: Example 26 • This says that the intersection of two
mutually exclusive events is the empty set
– I.e. these events have no outcomes in common
17 Visualizing Sample Spaces and
Events Using Venn Diagrams 18 Determining the Size of a Sample
Space
• The term counting techniques is used to refer to formulae
for computing the size of a sample space (or event)
• One such technique is the multiplication rule: Figure 28 Venn diagrams. 19 20 More Counting Techniques:
Permutations A Permutations Example Permutations: Permutations of subsets: Example 210 Permutations of subsets: 21 Further Permutations 22 Another Permutations Example Permutations of Similar Objects Permutations of Similar Objects: Example
211 23 24 More Counting Techniques:
Combinations Another Permutations Example
Permutations of Similar Objects: Example
212 Combinations 25 A Combinations Example 26 Interpretations of Probability
• Two philosophical approaches are
commonly taken: Combinations: Example 213 – Frequentist (aka objective):
• The relative frequency of occurrence, in a long run
,
yp
( g
p
of trials, of some type of event (e.g. a coin turns up
“heads”) – Subjective (aka Bayesian):
j
(
y
)
• The degree of belief in a statement, or the extent to
which it is supported by the available evidence (e.g.
the Calgary Flames will win the 2009 home opener)
27 28 Dealing With Discrete Sample
Spaces A Simple Frequentist Example
• 22.1 Introduction • For the special case where each member of
a sample space is equally likely to occur: • To determine the probability of an event:
Figure 210 Relative frequency of corrupted pulses sent
over a communications channel.
29 A Simple Example on Event
Probability (Example 2 15)
215 30 Another Example on Event
Probability
Example 216 Let th
L t the event E be that one of the 30 diodes meeting power
t b th t
f th
di d
ti
requirements is chosen, then P(E) is determined as follows: 31 32 The Axioms of Mathematical
Probability Addition Rules
The probability of the union of two events: The probabilit of the union of three e ents:
probability
nion
events: 33 A More General Definition of
Mutually Exclusive Events 34 An Example of Four Mutually
Exclusive Events • Slide #18 defined mutual exclusivity for two
events
• The concept generalizes to the case of k events as
follows (where 1 i k;
1 jk;
ij): Figure 212 Venn diagram of four mutually exclusive events
35 36 Visualizing This Example Using
a Venn Diagram Conditional Probability
• This concept deals with how the probability of some event
A should be reevaluated if we know that some other event
B has occurred
• Consider a manufacturing process example:
– 10% of items produced contain a visible surface flaw and 25% of
these are functionally defective
– 5% of the parts without a visible surface flaw are functionally
defective
– Let D denote the event that a part is defective and let F denote the
event that a part has a surface flaw
– Then we let P(DF) denote the conditional probability of D given
Then,
F, i.e. the probability that a part is defective, given that the part has
a surface flaw Figure 213 Conditional probabilities for parts with
surface flaws
37 Conditional Probability
Computations 38 A Conditional Probability
Example • The defining computation: Example 226 • The multiplication rule: 39 40 Using a Tree Diagram to Display
Conditional Probabilities Random Samples and
Conditional Probability
• To select from a batch randomly implies that at each step
of the sample the items that remain in the batch are
sample,
equally likely to be selected
• Example: • The data on 400 parts in the table below can be
visualized in a tree diagram:
i li d i
di – 50 parts in total, 10 from machine 1 and 40 from machine 2
p
y (without replacement) what’s the
p
)
– If 2 parts are selected randomly (
probability that the 1st is from m/c 1 and the 2nd from m/c 2?
– Let E1=first part selected is from machine 1; E2=2nd part selected is
from machine 2
– Since the sampling is random, P(E1)=10/50 and P(E2E1)=40/49
– Thus P(E1 ŀ E2) = P(E2E1) . P(E1) = 40/49 . 10/50 = 8/49
41 The Total Probability Rule 42 The Total Probability Rule(s)
Total Probability Rule (for two events): • This gives a way of determining the probability of an event
if enough conditional probabilities of the event are known Total Probability Rule (for multiple events): 43 44 A Total Probability Rule
Example Independence Example 227: semiconductor failure (and
contamination level): • The twoevent case: • The multipleevent case: 45 A Simple Independence Example 46 Bayes’ Theorem
• Preamble: rearrange eqn. 210 from earlier: • The theorem: 47 48 A Bayes’ Theorem Example (Ex. 237) Random Variables
Definition: Distinguishing random variables from real
experimental outcomes: 49 50 51 52 Types and Examples of Random
Variable
• Discrete v. continuous random variables: • Examples:
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 Fall '08
 NANAYAKKARA
 Probability theory

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