Statistics means never having to say you're certain.
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.
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.