There are liars, outliers, and out-and-out liars.
Chapter 6
Sec 6.2(b)
Some things to remember when assigning a probability to each individual outcome:
These probabilities must be numbers between 0 and 1 and must have a sum of 1.The
probability of any event is the sum of the probabilities of the outcomes making up the
event.
In some circumstances we assume that individual outcomes are equally likely such as
ordinary coins having a physical balance that should make heads and tails equally likely.
The table of random digits comes from a deliberate randomization making each number
equally likely.
There is a special rule for determining the probability of event A from a group of equally
likely outcomes is:
P(A) = outcomes in A
outcomes in S
OR
desired outcomes
total outcomes
Note: Most random phenomena do NOT have equally likely outcomes, so the general
rule for finite sample spaces is more important than the special rule for equally likely
outcomes.
Rule 4 (addition rule for disjoint events) describes the probability that one OR the other
of A and B will occur when A and B cannot occur together. Next we describe the
probability that BOTH events A and B occur in another special situation.
Independent events: two events A and B are independent if knowing that one occurs
does not change the probability that the other occurs. If A and B are independent, P( A
and B) = P(A)P(B). This is called the MULTIPLICATION RULE FOR
INDEPENDENT EVENTS. Independence is usually assumed as part of a probability
model when we want to describe random phenomena that seem to be physically unrelated
to each other. The multiplication rule extends to MORE than two events provided that all
are independent.
Repeating...the multiplication rule P(A and B) = P(A)∙P(B) holds only if A and B are
INDEPENDENT and not otherwise. The addition rule P(A or B) = P(A) + P(B) holds if
A and B are disjoint but not otherwise. So… disjoint events are not independent. A
Venn diagram can indicate disjoint but does not indicate independence.
If two events A and B are independent, then their complements Ac and Bc are also
independent and Ac is independent of B. By combining the rules we have learned that we
can compute probabilities for complex events. See Ex. 6.15 on page 354.
Probability thus far...
a) a random phenomenon has outcomes that we cannot predict but that do have a regular
distribution in many repetitions
b) the probability of an event is the proportion of times the event occurs in many
repeated trials
c) a probability model consists of a sample space S and an assignment of probabilities P
d) the sample space S is the set of all possible outcomes
e) P assigns a number P(A) to an event A
f) the complement of Ac consists of exactly the outcomes that are NOT in A
g) A and B are disjoint (mutually exclusive) IF they have no outcomes in common
h) A and B are independent if knowing one event occurs does not change the probability
of the other.
Five Rules so far...
1) All probabilities lie between 0 and 1.
2) P(S) = 1 (all outcomes added together have probability of 1)
3) COMPLEMENT RULE: for any event A, P(Ac) = 1 - P(A) from rule 2 and def. of
complement
4) ADDITION RULE: if A and B are DISJOINT then P(A or B) = P(A) + P(B).
(Union of events) -- If A and B are disjoint A AND B can NEVER occur together.
5) MULTIPLICATION RULE: if A and B are INDEPENDENT then (A and B) =
P(A)∙P(B) (Intersection of events).