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30 Cards in this Set
 Front
 Back
The probablity of an event is calculated by

summing the probabilities of the sample points in the sample space for A


Steps to calculating probabilities for mutually exclusive events
(5) 
1. define the experiment
2. list the sample points for each variable 3. assign probabilities to the sample points 4. define the set of sample points contained in the event of interest 5. sum the sample point probabilities to get the event probability 

Example of calculating probability: what is the probability of rolling at least a 9 with a single toss of two dice?
(5) 
1. The experiment is the single toss of 2 dice
2. The sample points are all the different values that the 2 dice can have. Interested in all that add up to 9, 10, 11, or 12 3. Probabilities of the sample points are: 1/36 4. the set of sample points contained in the event of interest are: (3,6) (4,6) (4,5) (5,6) (5,5) (5,4) (6,6) (6,5) (6,4) (6,3) 5. sum that sample point probabilities to get the event probability: a. there are 10 points in the event of interest, each having a probability of 1/36. therefore, the event probability is 10/36 = .28 

Compound event
(2) 
1. a composition of 2 or more events
2. can be the result of a union or an intersection 

Union

The union of two events A and B is the event that occurs if EITHER A or B or BOTH occur on a single performance of the experiment.


The union of events is written

A U B
A U B > all the sample points that belong to A or B or both 

Intersection

The intersection of two events A and B is the event that occurs if BOTH A and B occur on a single performance of the experiment.


The intersection of events is written

A (upside down U) B
A (upside down U) B consists of all the sample points belonging to BOTH A and B 

The Additive Rule of Probability

The probability of the UNION of events A and B is the sum of the probability of each event minus the probability of their INTERSECTION


Formula for the Additive Rule of Probability

P(A U B) = P(A) + P(B)  P(A udU B)


Mutually exclusive

If event A and event B are mutually exclusive, then their union, event C = A U B, looks like two separate circles


In other words, Events A and B are mutally exclusive if

A udU B contains no sample points


For mutually exclusive events,

P(A udU B) = 0


If B and C are mutually exclusive events,

their intersection is empty


Example of mutually exclusive events

Birthplace  there is nobody who was born in both Elko and Reno


Complement of an event

1. The complement of an event A, A^c, is the event that A does not occur.
2. A^c consists of all sample points that are not in A 

Example of complementary events

If A is at least 1 head in a toss of 2 coins, A^c is having no heads appear.
A: HH, HT, TH A^c: TT 

The Rule of complements

The sum of the probabilities of complementary events equals 1
P(A) + P(A^c) = 1 or P(A) = 1  P(A^c) 

Conditional Probability

1. Before, we assumed that all "conditions" were defined by the experiment.
2. But sometimes we have additional knowledge about conditions that can affect the outcome of an experiment. 3. You can think of this additional information as a reduction in the sample space that may affect the probability of some events 

Conditional Probability formula

P(A.line.B) = P(A udU B) / P(B)


The multiplicative rule and independent events

Events A and B are independent if the occurrence of one does not alter the probability of the other


If A and B are independent events, the formulas are:
(3) 
1. P(A.line.B) = P(A)
2. P(B.line.A) = P(B) 3. P(A udU B) = P(A) x P(B) 

Checking whether B and C are independent events
(2) 
Check whether
1. P(B.line.C) = P(B) 2. P(B udU C) = P(B) x P(C.line.B) 

Random sample
(df) 
n units are selected from a population in such a way that EVERY set of n units in that population has an equal probability of being selected


To find out how many random samples there are, we turn to

combinatorial mathematics


Combinations Rule

A sample of n elements is to be drawn from a set of N elements. The number of different samples possible is written:
(N over n) = N! / n!(Nn)! 

Example of combinations rule

Consider the task of choosing 2 marines from a platoon of 4 to send on a dangerous mission. How many different selections can be made?
Combinations Rule: (4 over 2) = 4! / (2!)(42)! = 4! / (2!)(2!) = 6 

Finding a random sample

The U.S. Bureau of the Census regularly samples the population for demographic information such as income, family size, employment, and marital status. Suppose the Bureau plans to sample 1,000 households in a town that has a total population of 53,432 households.
How could the Bureau use the random # table in Appendix A or a computer to generate the sample? Select the first 10 households in the sample. 

How could the Bureau use the random # table in Appendix A or a computer to generate the sample?
(5) 
my own words: bureau gives a # to every unit. then uses a computer to generate a sample of random #s
1. Number the households from 00,001 to 53,432 2. using the random number table, select a column at random, say column 8. 3. the first 10 numbers in column 8 will be the first 10 households to sample. 4. Select the next 990 between 00,001 and 53,432 (just discard the ones outside your range) 5. the sample will consist of the 1,000 households corresponding to the 1,000 different numbers 

To do random sampling of people in the household, use a

kish grid
(could take next birthday to get a random person to sample) 