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Transcript
Chapters 14-15
Elements of probability theory
• (probabilistic) trial
situation in which one of a collection of possible outcomes could occur, but precisely which one cannot be
predicted with certainty
• probability
long-run relative frequency of an event; probabilities are
estimated in three principal ways:
◦ empirically – by calculating the fraction of attempted
trials in which a desired outcome has occurred
◦ theoretically – by using a mathematical model to
describe the likelihood of occurrence
◦ subjectively – by making an “educated guess”
• Law of Large Numbers
The relative frequency of an event over a very long
string of trials approaches its true probability as the
number of trials grows ever larger.
1
Chapters 14-15
• disjoint events
events which have no outcomes in common, that is, can
never occur simultaneously
• independent events
events one of whose outcomes has no influence on the
outcomes of the other
2
Chapters 14-15
Formal rules of probability
1. Notation: The probability of event A is labeled P (A).
2. Probability measures likelihood : P (A) lies between 0
and 1 for any event A.
3. Something has to happen: Where S is the event consisting of the set of all possible outcomes, P (S) = 1.
4. Equally likely outcomes have equal probabilities: If there
are n equally likely possible outcomes and event A includes exactly k of these outcomes, then P (A) = k/n.
5. Complementary events have complementary probabilities: P (AC ) = 1 − P (A).
6. Addition rule for disjoint events: If A and B are disjoint
events, then
P (A or B) = P (A) + P (B).
7. Multiplication rule for independent events: If A and B
are independent events, then
P (A and B) = P (A) · P (B).
3
Chapters 14-15
More probability rules
• sample space
set of all possible outcomes of a probabilistic event
• Venn diagram
diagram of the sample space of an event (represented
by a rectangle) that depicts the relations among various
collections of outcomes (represented by circles which
might overlap)
• General Addition Rule
If A and B are any two events, then
P (A or B) = P (A) + P (B) − P (A and B).
4
Chapters 14-15
• conditional probability
If A and B are any two events, then the conditional
probability P (B|A) of “event B given event A” is the
frequency of the outcomes in B conditioned by the
outcomes in A; that is,
(rel.) freq. of outcomes in B also in A
P (B|A) =
(rel.) freq. of outcomes in A
This is equivalent to the formula:
P (A and B)
P (B|A) =
.
P (A)
• General Multiplication Rule
If A and B are any two events, then
P (A and B) = P (A) · P (B|A).
• Events A and B are independent precisely when their
conditional probabilities are the same as their unconditional probabilities; that is, when either one (and thus
both) of these formulas hold:
P (B|A) = P (B),
P (A|B) = P (A).
5
Chapters 14-15
• tree diagram
a diagram of the outcomes of pairs of successive events,
in which the first level of branches represent outcomes
of one event and the second layer outcomes of the second; useful for working with conditional probabilities
• Bayes’ Rule
Describes how to find the conditional probability of a
pair of events when the conditions are reversed: if we
know the conditional probability P (B|A), then we can
find P (A|B) using a tree diagram, or via Bayes formula:
P (A|B) =
P (B|A)P (A)
.
C
C
P (B|A)P (A) + P (B|A )P (A )
6