Chapter 4: Probability-Formulas
Chapter 4: Probability-Formulas
Combinations Rule
"different combinations" or "order doesn't matter"
# of combinations (order doesn't count)
when n diff. items are available, but only r of them
are selected w/o replacement
nCr=n!/(n-r)!r!
Permutations Rule (When some items are identical to others)
10 letters (aaaabbccde) are available and all
are selected, the # of diff. premutaions is:
10!/4!2!2!
# of diff. permutations when n items
are available and all n are selected w/o
replacement, but some of the items are identical
n!/(n_1 !n_2 !⋯n_k !)
Permutations Rule (When all items are DIFFERENT)
"If 5 letters are available, and
3 are to be selected w/o replacement"
# of diff. permutations when n diff.
items are available, but only r of them
are selected w/o replacement
nPr= n!
------
(n-r)!
Factorial Rule
"the # of ways five letters can be arranged"
5!:5*4*3*2*1
n!: # of diff. permutations (order counts)
of n diff. items when all n of them are selected
Factorial Counting Rule
"How many different characters are possible
if they're all represented"
Ex: 00110111: 2^8=720
m*n=# of ways 2 events can occur
Independent
Occurrence doesn't affect probability:
Sampling w/ replacement
Factorial symbol: !
Special: 0!=1
Example: 3!=3*2*1
Sampling w/o replacement
Depedent
Probability of event B occurring after
it is assumed that event A has already occurred
P(A and B)= P(A)*P(B) (independent)
P( A and B)=P(A)*P(B|A) (dependent)
Intuitive Addition Rule
Intuitive Addition Rule= P(A)+P(B)
---------------------------------------------
total # of outcomes in the sample space
Rule of Complementary Events
P(A)=1-P(Ā)
P(Ā)=1-P(A)
P(A)+P(Ā)=1
Formal Addition Rule
P(A or B)= P(A) + P(B) - P(A and B)
Probability of At Least One
4) Subtract the result from 1
P(at least 1 occurrence)= 1-P(no occurrences)
3) Find P(Ā)
2) Ā= getting none of the event
1) A= getting at least one of some event
Conditional Probability
P(B|A)= P(A and B)
------------
P(A)
"given that..."
Intuitive: P(B|A)
Equally likely outcomes
P(A)= # of ways A can occur
-------------------------
# of different simple events
Subjective Probability
P(A) is estimated by knowledge
of the relevant circumstances
Relative Frequency
P(A)= # of times that A occurred
------------------------------
# of times trial was repeated
Probability that event A DOESN'T occur
P(Ā)
Probability of event A
P(A)
Unusual: Extreme result
# of outcomes is far above/below typical values
Unlikely event
Probability= 0.05 or less
Multiplication
And= multiplication
P(A and B)
Addition
Or=addition
P(A or B)
Disjoint (Mutually Exclusive)
Two events cannot occur at the same time
