Statistics means never having to say you're certain.

Chapter 6

Sec 6.3

The purpose of learning more laws of probability is to be able to give
probability models for more complex random phenomena. We already have five
rules from the last section. Our addition rule only applies to DISJOINT
events. So...what happens when events "overlap" some???

**GENERAL RULE FOR UNION - ** (OR) for two events:

For ANY two events A and B, **P(A or B) = P(A) + P(B) - P(A and B)**

Note: When A and B are disjoint P(A and B) = 0 and the rule reverts to the
addition rule of before

Venn diagrams are a help in finding probabilities for unions, because you can
just think of adding and subtracting areas.

The simultaneous occurrence of two events A and B is called a JOINT event.
The probability of a joint event is called a JOINT PROBABILITY. Working
with joint events is best done with a table as described on pages 363 and 364.

**CONDITIONAL
PROBABILITY:**

is the probability assigned to an event when we KNOW some other event has
ALREADY occurred.

P(A|B) is the notation for conditional probability "read" probability of A given
that B has occurred or shorthand probability of A given B.

The formula for **P(A|B) = P(A and B) / P(B)** where P(B) > 0 which can be
manipulated to generate...

**GENERAL MULTIPLICATION RULE FOR ANY TWO EVENTS :**

**P(A and B) = P(A)P(B|A)**...in other words A __must__ occur followed by B

The UNION of a
collection of events is the event that ANY of them occur.

The INTERSECTION of any collection of events is the event that ALL of them
occur.

To extend the multiplication rule to the probability that all of several events
occur, the key is to condition each event on the occurrence of all of the
preceding events, like stacking....for events A, B, and C this looks like

P(A and B and C) = P(A)P(B|A) P(C|A and B)

As problems get more complex, drawing "tree diagrams" can be extremely useful to
follow all the "what ifs".

See page 373. Tree diagrams combine the addition and multiplication rules.
The multiplication rule says that the probability of reaching the end of any
complete branch is the product of the probabilities written on its segments.
The probability of any outcome is found by adding the probabilities of all
branches that are part of that event.

True understanding of the forward probabilities in a tree diagram will insure
the ability to "reverse" the process.

Thomas Bayes (Baye's rule) created formulas for this revere process and if you
are interested in memorizing more "stuff" it can be found on page 375.

P(B) is NOT usually equal to P(B|A) since sometimes knowing that A has occurred
provides some additional information about whether or not event B can occur.
This gives rise to the definition of independent. Two events A and B are
INDEPENDENT if P(B|A) = P(B).

Decision analysis is decision making in the presence of *uncertainty* that seeks
to make the probability of a favorable outcome as large as possible. This
is the type of decision making made in business by managers and other
responsible for major decision making.