CHAPTER 5 PROBABILITY
CHAPTER 5PROBABILITY
CARDS & DICE
BLACK RED
CLUB SPADE DIAMOND
HEART
TOTAL
ACE 1 1 1 1 4
FACE CARD (K, Q, J)
3 3 3 3 12
NUMBERED CARD (1-9)
9 9 9 9 36
TOTAL 13 13 13 13 52
CARDS & DICE Single Die: Six sides Numbered 1 to
6
Double Dice: Use the sum of the two top sides (totals are 2 to 12)
PROBABILITY DEFINITIONS: Probability – The likelihood that a given event
will occur. Expressed as a proportion. Experiment – A procedure that can be
repeated that has uncertain random results. Sample Space – All possible outcomes of an
experiment or observation – Examples. Event – Subset of the Sample Space (could be
simple or complex). Notation – P(A) = Probability of A occurring.
TWO WAYS TO DETERMINE: Empirical – Trial;
Run a number of times and count successes; Then
Classical –
Calculated based on known outcomes
Examples of each.
#_ _( )
#_ _
OF SUCCESSESP A
OF TRIALS
#_ _ _ _ _( )
#_ _ _ _
OF WAYS A CAN OCCURP A
OF ALL POSSIBLE OUTCOMES
LAW OF LARGE NUMBERS
As the number of trials increases using the Empirical Method, the closer the trial P(A) approaches the Classical (actual) P(A).
PROBABILITY PROPERTIES The probability of an impossible event is
0.0.
The probability of an absolutely certain event is 1.0.
For any event A, 0.0 ≤P(A) ≤ 1.0.
The sum of probabilities of all events in the Sample Space is 1.0.
PROBABILITY DEFINITIONS:
Probability Model: A table listing each outcome of the Sample Space and its Probability of Occurrence.
For the Table to be a true Probability model, all outcomes must be listed; all the probabilities must meet probability definition; and the sum of all probabilites must add to 1.0.
PROBABILITY DEFINITIONS:
Unusual Event: Any event whose probability is less than 0.05 or 5%.
PROBABILITY EXAMPLES
For each, show sample space as well as probabilities:
Coins
Jar of Colored Marbles
Triplets
Cards
Dice – Do full Probability Space
DEFINE DISJOINT (5.2): Disjoint (Mutually Exclusive) Events –
Events that can not occur simultaneous.
Venn Diagram
Examples of Mutually Exclusive Events Flipping a coin. Drawing a card. Picking a marble from a jar.
0P A B
ADDITION (OR) RULE: If Events A & B are Mutually Exclusive
(or disjoint), the Probability of Event A or Event B occurring is P(A) + P(B).
( ) ( )P A B P A P B
0P A B
DEFINE NOT DISJOINT (Not Mutually Exclusive):
Examples of events that are Not Disjoint: Cards of number and suit. Marbles of colors and letters.
Venn Diagram 0P A B
ADDITION (OR) RULE: If Events A & B are NOT Mutually Exclusive (or
disjoint), the Probability of Event A or Event B occurring is P(A) + P(B) – the overlap P(A and B).
AND
Venn Diagram
Use with given probabilities.
( ) ( )P A B P A P B P A B
0P A B
ADDITION (OR) RULE:
Examples of disjoint probabilities
Cards
Marbles
Tables
ADDITION (OR ) RULE:
FRESHMEN SOPHOMORE JUNIOR SENIOR TOTAL
SATISFIED 57 49 64 61 231
NEUTRAL 23 15 16 11 65
NOT SATISFIED 21 18 14 26 79
TOTAL 101 82 94 98 375
COMPLIMENT RULE: Probability of A not occurring is P A
1
1
P A P A
then
P A P A
COMPLIMENT RULE: If A, B, C, D & E are all possible events,
then
And
( ) 1P A P B P C P D P F
( ) 1P A P B P C P D P F
DEFINE INDEPENDENCE(5.3): Two events are independent if the
occurrence of one event does not affect the probability of the occurrence of the other event.
In other words, P(B) is the same whether or not event A has occurred or not.
Occurrence of event A does not affect probability of event B.
MULTIPLICATION (AND) RULE:
If two events are independent then
Examples:
Die
Marbles with replacement.
Games
*P A B P A P B
CONDITIONAL PROBABILITY(5.4):
Means the probability of B occurring GIVEN THAT event A has already occurred.
The statement “Given That” CHANGES THE SAMPLE SPACE.
FORMULA:
Do with given probabilities.
|P B A
( )( | )
( )
P A BP A B
P A
CONDITIONAL PROBABILITY:
FRESHMEN SOPHOMORE JUNIOR SENIOR TOTAL
SATISFIED 57 49 64 61 231
NEUTRAL 23 15 16 11 65
NOT SATISFIED 21 18 14 26 79
TOTAL 101 82 94 98 375
MULTIPLICATION RULE FOR EVENTS THAT ARE NOT INDEPENDENT: If events A & B are NOT independent
Examples
Marbles without replacement
Cards without replacement
Table
* |P A B P A P B A
MULTIPLICATION RULE FOR LARGE POPULATION:
If the population is very large (size of a large town) then we can consider two events as independent even without replacement (consider it as with replacement).
Example of survey.
ACCEPTANCE SAMPLING EXAMPLE: Lot of 100 circuits has 5 defective. Take 2 circuits without replacement. If
only one defective then reject the lot. What is the probability of rejecting the
lot? Build tree of good/bad using conditional
probability. Use the addition & multiplication rules
to find probability of lot being rejected.
ACCEPTANCE SAMPLING EXAMPLE:
Pass 94/99 2nd Sample95/100 Pass Fail 5/99 1st Sample 5/100 Fail Pass 95/99 2nd Sample
Fail 4/99
PROBABILITY SUMMARY
Good Summary of Probability Rules on Page 284 of text.
There is NO relationship between Mutually Exclusive and Independence concepts.
COUNTING METHODS:
Using the classical method of calculating probabilities, we need to find better ways to count possibilities.
Example of births of triplets.
COUNTING METHODS: Multiplication Rule with Replacement:
How many three or five digit numbers? How many “word” combinations with 6
letters? How many meals?
Formula: n items to select from and want to select r items -
rn
COUNTING METHODS: Multiplication Rule without
Replacement: Define Factorial.
How many three digit numbers using only numbers 0, 1, 2 without replacement?
How many word combinations with 6 letters without replacement?
How many ways to arrange 5 books without replacement?
COUNTING METHODS: Multiplication Rule without
Replacement and with more items than options:
Have n items but only r places for them.
Order matters: PERMUTATION 8 books but only 5 places 10 people 5 offices
Formula:
!
!n r
np
n r
COUNTING METHODS: Multiplication Rule without Replacement
and with more items than options: Have n items but only r places for them. Order does not matter: COMBINATION
8 books but only 5 places 10 people committee of 5 Lottery
Formula: !
! !n r
nC
r n r
COUNTING METHODS:
CALCULATOR FUNCTIONS
COUNTING METHODS: Arrange n items n ways, if not all n
items are distinct. Formula:
Where the are non distinct items.
Example using fruits or names.
1 2 3
!
* * *....* k
n
n n n n
'in s
COUNTING METHODS:
SUMMARY ON PAGE 304READ SECTION 5.6
PROBABILITY QUOTES “The 50-50-90 rule: Anytime you have a 50-50
chance of getting something right, there's a 90% probability you'll get it wrong.” Andy Rooney
“From principles is derived probability, but truth or certainty is obtained only from facts.” Tom Stoppard
“Life is a school of probability.” Walter Bagehot
PROBABILITY