Probability

3.1 Events, Sample Spaces, and ProbabilitySample space - The set of all possible

outcomes for an experiment

Roll a die

Flip a coin

Measure heights

}6,5,4,3,2,1{space sample

}TH,{space sample

tallest}ofheight person tosmallest of{height space sample

Some experiments consist of a series of operations. A device called a tree diagram is useful for determining the sample space.

ExampleFlip a Penny, Nickel, and a Dime

Event - Any subset of the sample space

An event is said to occur when any outcome in the event occurs

The probability of an event A, denoted , is the expected proportion of occurrences of A if the experiment were performed a large number of times.

When outcomes are equally likelyExamples: Flip a fair coin

Roll a balanced die

)(AP

outcomes ofnumber Totalevent tofavorable outcomes ofNumber eventan ofy Probabilit

When probability is based on frequencies

ExampleResults of sample

Males (event M) – 40Females (event F) – 60

n size Sampleevent ofFrequency eventan ofy Probabilit

The closer to 1 a probability the more likely the event

1)(0 AP

1space) sample( P

3.2 Unions and IntersectionsJoint Probability – an event with two or more

characteristics

The union of two events, denoted , is the event composed of outcomes from A or B. In other words, if A occurs, B occurs, or both A and B occur, then it is said that occurred.

The intersection of two events, denoted , is the event composed of outcomes from A and B. In other words, if both A and B occur, then it is said that occurred.

BA

BA

BA

BA

3.3 Complementary Events

The complement of an event A, denoted , ,

or , is all sample points not in A.

The complement rule:

)(AP

)(AP

)(1)( APAP

)( cAP

3.4 The Additive Rule andMutually Exclusive EventsThe addition rule

We say the events A and B are mutually exclusive or disjoint if they cannot occur together

0) ( BAP

) ()()() ( BAPBPAPBAP

3.5 Conditional ProbabilitySometimes we wish to know if event A

occurred given that we know that event B occurred. This is known as conditional probability, denoted A|B.

The conditional probability rule for A given B is

)() ()|(

BPBAPBAP

Red Die

Green Die1 2 3 4 5 6

1 (1,1) (2,1) (3,1) (4,1) (5,1) (6,1)2 (1,2) (2,2) (3,2) (4,2) (5,2) (6,2)3 (1,3) (2,3) (3,3) (4,3) (5,3) (6,3)4 (1,4) (2,4) (3,4) (4,4) (5,4) (6,4)5 (1,5) (2,5) (3,5) (4,5) (5,5) (6,5)6 (1,6) (2,6) (3,6) (4,6) (5,6) (6,6)

ExampleRoll a balanced green die and a balanced red dieDenote outcomes by (G,R)}7 is dice theof sum{A

}4 numbersboth { B}1 is diegreen {C

ExampleSelect an individual at random from a population of drivers classified by gender and number of traffic tickets

0 tickets 1 ticket 2 tickets 3 or more tickets TotalFemale 1192 321 72 15 1600Male 695 487 141 77 1400Total 1887 808 213 92 3000

}female isdriver selected{A

} tickets2least at hasdriver selected{B

3.6 The Multiplicative Rule and Independent EventsTwo events are said to be independent if the

occurrence (or nonoccurrence) of one does not effect the probability of occurrence of the other.

Events that are not independent are dependent.

)|()( BAPAP

)|()( BAPAP

ExampleDraw two cards without replacement

Multiplication rule

Suppose we return the first card and thoroughly shuffle the cards before we draw the second

ace}an is cardfirst {A}acean is card second{B

)|()() ( ABPAPBAP

ExampleSelect an individual at randomAsk place of residence andDo you favor combining city and county

governmentsFavor (F) Oppose Total

City (C) 80 40 120Outside 20 10 30Total 100 50 150

3.7 Random SamplingA simple random sample of n measurements from a population is one selected in such a manner that every sample of size n from the population has equal probability of being selected, and every member of the population has equal probability of being included in the sample.

3.8 Some Additional Counting RulesHow many different ways are there to arrange

the 6 letters in the word SUNDAY?

Suppose you have a lock with a three digit code. Each digit is a number 0 through 9. How many possible codes are there?

The symbol, read as “n factorial” is defined as

and so on

!n

1!0 1!1

212!2 6123!3

241234!4

Evaluate each expression

!2!5

!8!9

!6!2!8

PermutationsOrdered arrangements of distinct objects are called

permutations. (order matters)

If we wish to know the number of r permutations of n distinct objects, it is denoted as

In how many ways can you select a president, vice president, treasurer, and secretary from a group of 10?

)!(!rnnPrn

CombinationsUnordered selections of distinct objects are

called combinations. (order does not matter)

If we wish to know the number of r combinations of n distinct objects, it is denoted as

In how many ways can a committee of 5 senators be selected from a group of 8 senators?

)!(!!rnr

nCrn