Chapter 4 Notes

©2009 by S. Gramlich

Probability

! = Important Note

! These Notes are not meant to replace Reading.  Read Chapter first.

 

4-2

Event = an outcome

Sample Space = all possible events

Probability = chance of an event happening (expressed either as a fraction, decimal, or %)

Probability of an Event A:

P(A) = s/n

            s = # of ways A can or has occurred

            n = sample size = total # of possible ways (count # events in sample space)

 

0 <= P(A) <= 1

complement of A = not A (Abar or A'):  P(Abar)= 1 - P(A)

 

4-3

mutually exclusive (disjoint)= events cannot occur at same time (exclusive)

Addition Rule:  P(A or B) = P(A) + P(B) - P(A ∩ B)

P(A ∩ B) = A and B at the same time

 

Contingency Table = Table with column and row variables

! add up the column and row totals first,

the sum of row totals = the sum of column totals = Grand Total = N = Sample Size

 

4-4

Differentiate between Independence and Dependence?

            Independent = events aren't affected by each other, sampling done with replacement

            Dependent = events are affected by each other, sampling done without replacement

 

Prob of A and THEN B:

            P(A and B) = P(A)*P(B) if independent

            P(A and B) = P(A)*P(B\A) if dependent

 

! use the s/n formula for calculating both events

P(B\A) = Prob of B occurring, given A has already occurred;

the sample size for calculating B will be 1 less than what was used for A if dependent

 

4-5

Prob (at least 1) = 1 – P(none)

P(X >= 1) = 1 – P(0)

 

Conditional Probability P(B\A) = P(A ∩ B)/ P(A)

 

4-6 Simulation

Law of Large Numbers (from 4-2): as procedure is repeated again and again, relative frequency probability (simulation) will approach actual probability

 

4-7 Counting= finding N of P(A) = S/N

Multiplication rule (independence, with replacement):

N = # of ways selection 1 * # of ways selection 2 * # of ways selection 3 *…..* # of ways selection k

 

Factorial rule (dependence, without replacement):

N = x! = x(x-1)(x-2)*…*1

 

Permutation rule:

Select r of n items without replacement

Order matters (ab not equal to ba)

(Like selecting a portion of items from factorial rule)

N = nPr = n! / (n –r)!

 

Combination rule:

Select r of n items without replacement

Order doesn’t matter (ab = ba) so will have less total ways than a Permutation

(Like selecting a portion of items from factorial rule)

N = nCr = n! / [(n –r)!r!]

 

4-8 Bayes Thereom

P(A\B) = [P(A) * P(B\A)] / [P(A)*P(B\A) + P(notA)*P(B\notA)]

!note that numerator and 1st half of denominator are the same value

 

 

TECHNOLOGY

Excel commands:

=fact( )

=permut( )

= combin( )