additional 2 Flashcards
What is probability
a numerical measure of the likelihood that an event will occur
What can probability be used for
used as measures of the degree of uncertainty
probability values are always assigned on a scale of
0 to 1
what is a prob of near zero indicate
event is unlikely to occur
what is prob of near 1 indicate
an event is almost certain to occur
what is an experiment
a process that generates well-defined outcomes
what is a sample space
set of all experimental outcomes
what is an experimental outcome also called
a sample point
what is a sample point
identifies an element of the sample space
What is the counting rule for a multistep experiment
if an experiment can be described as a sequence of k steps with n1 possible outcomes on the first step, n2 possible outcomes on the second step, and so on
What is a tree diagram
a graphical representation that helps us visualize the multi-step experiment
In a tree diagram, along the branches you
multiply
What are the 3 useful Counting Rules
- Multistep experiment
- Combinations
- Permutations
What are combinations
- allows us to count the # of experimental outcomes when the experiment involves select n objects for a set of N objects and ORDER DOES NOT MATTER
What are permutations
- allows us to compute the # of experimental outcomes when n objects are to be selected from a set of N objects and ORDER MATTERS
Does permutations or combinations result in more
permutations
b/c every selection of n objects can be ordered in n! different ways
What are the 3 most useful approaches for assigning probabilities
- Classical Method
- Relative Frequency method
- Subjective method
Explain classical method
use when: all experimental outcome are EQUALLY LIKELY
- the 2 basic requirements are satisfied
- example tossing a coin
Explain when to use the Relative Freq. Method
use when: data re available to estimate PROPROTION Of TIME and the experimental outcome will occur if the experiment is REPEATED a LARGE # of TIMES
- 2 basic requirements are satisfied
Explain when you would use the Subjective Method
Use When: we CANNOT realistically assume that the experimental outcomes are Equally Likely and when Little relevant data is available - degree of belief
Which method would we use Bayes Theorem for
Subjective method
what are the basic requirements for assigning Probabilities
- the prob assigned to each experimental outcome must be b/w 0 and 1 inclusively
- Sum of prob for all experimental outcomes must equal 1.0
What is an event
collection of sample points
what is the prob of an event
equal to the sum of the prob(s) of the sample points in the event
Even in business situations where either the classical or relative freq. method is used, managers may want to provide subjective prob estimates. In such cases, the best prob estimates often are obtained by combining the estimates from the classical or relative frequency with the
subjective method
Prob of A OR B is shown as and what rule does it use
P(AUB) A Union B - uses addition rule
P(A) + P(B) - P(ANB)
If A and B are mutually exclusive the how do you compute the Prob of A OR B
P(AUB) = P(A) + P(B) - 0
How do you compute the prob of A AND B
P(AnB) = P(A) x P(B)
Dependent: P(AnB) = P(A)+P(B) - P(AUB)?
If events are Independent then the P(A/B) =
P(A)
When do you use the addition law P(AUB)
knowing the prob of at least one of two events occurring
- prob that event A OR event B OR Both occur
Event A and B is the event containing
all sample points belong to A or B or Both
What is the intersection of two events
the event containing the sample points belonging to BOTH A and B denoted by A n B
P(A) = .3, P(B) = .7 P(A|B) = .40 are they independent or dependent
they are dependent b/c P(A|B) does not equal P(A)
P(A) = .3 P(B) = .4 and P(ANB) = .12 are they independent or dependent
they are independent because P(A) x P(B) = .12
The multiplication law is used to compute the
intersection of 2 events
The multiplication law is based on the definition of
conditional probability, need to know if it is dependent or independent
With Multiplication law P (ANB) the item has replacement then it is
independent (b/c the prob of being chosen is the same every time)
with multiplication law P(ANB) the item is not replaced then it is
Dependent (b/c chances change with each selection
When do you use Bayes Theorem
when revising probabilities when new info is obtained
(used subject method prior)
- for computing posterior probabilities
What is a random variable
a numerical description of the outcome of an experiment
- must assume numerical values
what types of random variables are there
- Discrete
2. Continuous
Describe a discrete random variable
finite
# of books, age, can count them
values of f(x) must be greater than or equal to 0
Describe a continuous random variable
- infinite
- glass of water, temperature, size, weight, distance, time
- collection of intervals
What does a prob distribution for a random variable describe
describes how prob(s) are distributed over the values of the random variable
what are the 3 methods used for discrete random variables
- relative frequency method
- classical method
- subjective method
When do you use the Relative Frequency method
- when reasonably large amounts of data are available
- treats the data as though it was a population
- more widely used in practice
- leads to what is called an Empirical Discrete distribution
When do you use the classical method
- when experimental outcomes generate values of the random variable that are equally likely
What are the required conditions for the discrete prob function f(x)
F(x) must be greater than or equal to 0
Sum f(x) =1
What is the primary advantage of defining a random variable and its prob distribution
once the prob distribution is known, it is realtively easy to determine the prob of a variety of events
what are the 4 properties of a binomial experiment
- experiment consists of n identical trails (fixed number of trials)
- 2 outcomes are possible on each trial (success or failure)
- The prob of a success is denoted by p Does not change from trial to trial
the prob of failure is denoted by 1-p does not change from trial to trial - the trials are independent
What are the properties of a Bernoulli process
- 2 outcomes are possible
- prob of success is denoted by p, failure 1-p
- trails are independent (prob does not change)
What is a possion probability distribution
often used to model random arrivals in waiting line situations,
- useful in estimating the # of OCCURRENCES over a SPECIFIED INTERVAL of time or space
What are the properties of a poisson distribution
- the prob of an OCCURRENCE is the same for any 2 intervals of equal length
- the occurrence or nonoccurrence in ANY interval is independent of the occurrence or non-occurrence in any other interval
When do you use the Hypergeometric
when
- the trials are not independent (no replacement)
- prob of success changes from trial to trail
for hypergeometric r is
the number of elements in the pop labelled as successes
Hypergeometric N-R is
the number of elements in the pop labelled failure
Hypergeometric N is
number of elements in the population
Hypergeometric x is
prob of success we are looking for
Hypergeometric n is
number of trials
