CHAPTER 5 SUMMARY
Probability Rules:
Probability of an Outcome A (or Event A) – denoted as P(A) – is the likelihood
that A will occur. Sample Space is the set of all possible outcomes.
The probability of an event that could not possibly occur is 0 and the probability
that an event will certainly occur is 1. So for any probability 0 ≤ P(a) ≤ 1. The
sum of the probabilities of all possible outcomes in the sample space is 1.
Two methods of finding a probability of an event:
Frequency ( A)
where n is the number of trials.
n
#_ of _ ways _ A _ can _ occur
Classical – P( A) 
.
#_ of _ all _ possible _ outcomes
Empirical (or Experimental) – P( A) 
The empirical method is an estimate of the true probability which is the determined
with the classical method.
Law of Large Numbers – As the number of trials (n) increases P(A) from the
empirical method approaches the P(A) of the classical method.
Mutually Exclusive Events and the Addition Rule and Compliments:
If two events, A and B, can NOT occur at the same time they are said to be
Mutually Exclusive (or Disjoint).
If two events, A and B, are mutually exclusive then P  A B   P( A)  P( B) .
If two events, A and B, are NOT mutually exclusive then
P  A B   P( A)  P( B)  P  A B .
 
 
P A  1  P( A) where P A means the probability of A not occurring.
Independence and the Multiplication Rule and Conditional Probability:
Events A & B are said to be Independent if the occurrence of A does not effect the
probability of B occurring.
P  B | A is the probability of B occurring given that A has already occurred.
If two events, A and B, are independent then P  A B   P( A)* P( B) .
If two events, A and B, are NOT independent then P  A B   P( A)* P  B | A .
Counting Events:
Multiplication Rule: If there are several choices to make (p being the number of
the first choices, q being the number of second choices and r being the number of
third choices) then the total number of all choices is p*q*r.
Counting With Replacement: If you are selecting from n items and you can reuse
the items, then you have n choices for the first selection. Since you can reuse the
item you have n choices for the second choice. The same is true for all subsequent
choices. If you are making a total of r choices, then the total number of ways r
items can be selected from n items is n r .
Counting Without Replacement: If you have n choices for the first event but can
not reuse that choice, then you only have n-1 choices for the second event and n-2
for the third, and so on. If you are selecting n choices from n items then the total
ways they can be selected is n*(n-1)*(n-2)*……..*2*1 or n!.
Permutations: If you are making r choices from n items and you can not replace
items and the order of the items selected matters, then the total ways r items can be
selected is denoted as n Pr which is
n!
.
(n  r )!
Combinations: If you are making r choices from n items and you can not replace
items and the order of the items selected does not matter, then the total ways r
items can be selected is denoted as n Cr which is
n!
.
(n  r )!r !