Definitions, Set Theory and Counting

Chapter 2: Probability

Classical DefinitionIf a random event can occur in n equally

likely and mutually exclusive ways, and if na

of these ways have an attribute A, then the probability of occurrence of the event having attribute A is written as:

n

nAobPr a)(

Relative Frequency DefinitionIf a random event occurs a large number of

times n and the event has attribute A in na of these occurrences, then the probability of the event having attribute A is

n

nAprob a

n lim)(

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1895 1915 1935 1955 1975 1995

year

Prob

abili

ty

Prob(peak > mean)

A

SE1

E2

E4E3

1)(0 iEprobiiES 1)()( i iEprobSprob

inmi EA

n

mi iEprobAprob 1)()(0

AB

S E1E2

E4E3

BA

BA

)()()()( BAprobBprobAprobBAprob

A

B

S E1E2

E4E3

0)( BAprob

)()()( BprobAprobBAprob

Independent Events vs. Mutually Exclusive Events Independent events: If the occurrence or

nonoccurrence of an event A, has no relation to or influence on the occurrence or nonoccurrence of an event B (and vice versa) then the two events are independent and

If A and B have no outcomes in common then A and B are mutually exclusive and

)()()( BprobAprobBAprob

0)( BAprob

S

A

E1E2

E4E3

SAA c 1)()()( cc AprobAprobAAprob

)(1)( cAprobAprob

Ac

Conditional ProbabilitiesIf the probability of an event B depends on

the occurrence of an event A, then we write prob(B|A), the probability of B given A has occurred OR the conditional probability of B given A has occurred.

A

B

BA

0)()(

)()|( Aprob

Aprob

BAprobABprob

)|()()( ABprobAprobBAprob

Conditional Probability of Independent Events

)()(

)()(

)(

)()|( Bprob

Aprob

BprobAprob

Aprob

BAprobABprob

Total Probability TheoremIf B1, B2, …Bn

represent a set of mutually exclusive and collectively exhaustive events, one can determine the probability of another event A from

)()|()(1 i

n

i i BprobBAprobAprob

B1

B2 B3

B4

B5B6

A

S

Example: Total Probability TheoremExperimentally we have found that in May

the probability of the water temperature in the Navasota River reaching over 50°F when the river is flowing at a rate greater than 3000 cfs is 0.40. The probability of the temperature being over 50°F when the river flowing at a rate less than 3000 cfs is 0.70. The probability on any given day in May that the river will exceed 3000 cfs is 0.65. What is the probability that the river will exceed 50°F on any random day in May?

SamplingSelecting r items from n items.Can be done in one of two ways

With replacementWithout replacement

4 types of samplesOrdered with replacementOrdered without replacementUnordered with replacementUnordered without replacement

Sampling-Ordered with replacementThe number of ways of selecting r items

from n items with replacement and order is important is nr.

Example: 3 blocks labeled 1 to 3. How many ways can we draw 2 of the 3 blocks when order matters and the blocks are replaced each time?

Sampling: Ordered without replacement

r ordered items can be selected from n without replacement in n(n-1)(n-2)…(n-r+1) ways.

Commonly written as (n)r and called the number of permutations of n items taken r at a time.

)!(

!)1)...(2)(1()(

rn

nrnnnnn r

)1)(2)...(2)(1(! nnnn

Sampling without replacement, unordered (i.e. order does not matter).

Similar to ordered sampling without replacement except that in ordered sampling the r items selected can be arranged in r! ways. Thus r! of the ordered samples contain the same elements.

The number of different unordered samples will then be commonly written as

and called the binomial coefficient . Gives the number of combinations of

selecting r items from n without replacement.

!

)(

r

n r

r

n

!)!(

!

! rrn

n

r

n

r

n r

Unordered sampling with replacement

Selecting r items from n with replacement when order doesn’t matter is equivalent to selecting r items from n+r-1 items without replacement.

We can consider the population as containing r-1 more items than it really does since the items selected will be replaced.

!)!1(

!11

rn

rn

r

rn

Example: Sampling Shortly after being put into service some

buses manufactured by a certain company have developed cracks on the underside of the main frame. Suppose that a particular city has 20 of these buses, and cracks have actually appeared in eight of them.

1. How many ways are there to select a sample of five buses from the 20 for a thorough inspection?

2. In how many ways can a sample of five buses contain exactly five with visible cracks?

3. If a sample of five buses is chosen at random, what is the probability that at least four of the five has visible cracks?