math 2

probability

likelihood or chance of an event occurring

not likely event

probability is less than or equal to 0.05

event

collection of results or outcomes of a procedure

simple event

outcome or an event that cannot be further broken down into simpler components

sample space

consists of all possible simple events. the sample consists of all outcomes that cannot be broken down any further. has to be written order included how many possibilities total

a, b, c

denotes a specific event

p(A)

denotes the probability of event A occurring

the probablity of an event is a

fraction or decimal between 0 and 1 percentage 0% and 100%

relative frequency approximation of probability

conduct or observe a procedure and count the number of times that event A occurs. P(A) is then solve as follows

P (A) = # of times that event A occurs

over\

# of times procdure was repeated

Law of Large Numbers

a procedure is repeated again and again, the relative frequency probability of an event tends to a approach the actual proabality

classical approach to proabality

this requires equally likely outcomes. if a procedure has n different simple evdnts that are equally likely, and if event A can occur different wats the s - total events occuring\\

P s over m

Subject proablity

: P(A), the probability that event A is estimated by using knowledge of relevant circumstances. (Basically an eductaed guess

complementary events

Sometimes we need to find the probability that an event A does not occur. Thus, we need the __complement ______________. The complement of event A, denoted by _______a bar_____, consist of all outcomes in which event A ___does not _________ _____ occur.

interpreting probabilities

If the probability of an event is _very small (less than or equal to 0.5 _______ then the event is __unlikely______________ to happen. If the event has a very small probability of occurring and does occur, this is considered an __unusual_______________ event

a compound event

any event combining two or more simple events

addition rule

Notation : P (A or B) the probalilbity that A occures or B occurs or both occur in a single trial

The formal addition rule

P(A or B) = P(a) + p(b) -p (a and b), where p( a and b) is the proablity that A and B occur at the same time in the same trial

events A and b are disjoint or mutually exclusive if

they cannot occur at the same time in the same trial

mutatally exclsuive sepereate events cant happen together

if A and B are disjoint then

P (A and B) = 0

complementary events and the addition rule

since complementray events are always disjoint, the following rules are true:

P (a or A complement) =P (A) + P ( a compkement) = 1

multiplicatuon rule

see the word and

muiltiplication rule

notation: P(A and B) the probalivy that event A occurs and event B occurs; P( event A occurs in a first trial and event B occurs in a second trial)

In the multiplication rule p( A and B) has a different meaning than above

independent events

thw occurence of one event does not affect the probalvity of the occururnce of the other

if two events A and B are independent

the occurnece of one does not affect the proablity of the occurence of the other. So, if A and B are indepenet events, P(A and B)= P(A) times p(B)

sam[ling with replacment

indepentent - put back into

if two events are dependent

p (A and B) = P( A) times P(B/ A)

event B occurs given that A has already happened

Exception

treating dependent events as independent

this exception is for when a small sample is drawn from a large population. it would be rare to select the same item twice. the common guideline is when sampling without replacement and the sample size is no more than 5% of the population treat the selections as being independent (even though dependent)

random variable

variable that has a single numerical value determined by chance for each outcome of a procedure

discrete random variable

has a collection of values thatb is finite or countable

ex. given a test with 50 multiple choice questions random variable is the number question correct

continous random variable

infinietly many values and the collection of values is not countable

proablity distrubuition

a description that gives the probablity for each value of the random variable- can be tables, graphs, or formulas.

3 requirements to have a probablity distrbuition

there is a numerical random varisable and its values are assiociated with proablities

sum of proablities( x) =1 where x assumes all possible values of random variable

0 is less than or equal p (x) is greater than or equal to 1. for every value of the random variable x.

population mean (m)

central value of the random caraible and is called the expected value

the population standard deviation

measures the variation of the rabdom variable about the mean.

round the mean and standard dev one decisml place past te orgunal data values

p (x or more) less than or equal ro 0.05 the x is consdiered

sign. hugh number of successes

if p( x or less) lless than or equal to 0.05

considered sign low number of success

a binomial proablity distribution

resulets from a prodecure that a fixed number of trails( A trial is a singke obersevatiob) This fixed number of trials is dneoted by n

the r]trials must be indeoenenet

each trial must have all outocmes classidef int two categoresis or outcomes

a. sucess/ failure’

b. p (sucess) =p

c. p (failure) = q

d. p + q = 1

the proablity of a success must remain the same in all trials

n

fixed number of trials

P (success)

p

k

number of success out on n trials

proablity of getting exactly k successes out on n trials

P( x <k) = binomals pdf (n,p,k)

proablity of getting at most k successes out on n trials

p (x<k)= binomal cdf (n,p,k)

proablity of getting at least k succesees out of n trials

P( x>k)= 1- binalcdf (n,p,k-1)

binoaml mean expectec value

m= np

binoaml standard dev

square rooot of n p q

sign hugh number of successses

x success among n trials is sign high if the proablity of x or more is succsess is 0.05 or less

p(x or more) < or equal to 0.05

sign low number of successes

x succdess a ong n trials s sign low if the proablity of x or less is 0.05 or less