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