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 sum 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 like 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 in
English 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 indicated disjoint
but does not indicate independence.

If two events A and B are independent, then their complements A^{c}
and B^{c} are also independent and A^{c} 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) sets of outcomes are events

f) P assigns a number P(A) to an event A

g) the complement of A^{c }consists of exactly the outcomes that are NOT
in A

h) A and B are disjoint (mutually exclusive) IF they have no outcomes in common

i) 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(A^{c}) = 1 - P(A)
from rule 2 and defn 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)

(interesection of events).

Index