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Level 225

Probability II


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probability
is measured between 0 and 1
experimental probability
What the outcomes did turn out to be in an experiment.
Theoretical Probability
What the outcomes were supposed to be theoretically.
complement of an event
all the possible outcomes in the sample space that are not part of the event
probability of a complement
the sum of the probability of an event and the probability of its complement is 1
frequency table
a data display that shows how often an item appears in a category
Relative Frequency
the ratio of the frequency of a category to the total frequency
probability distribution
shows the probability of each possible outcome
Fundamental Counting Principle
(# of choices)(# of choices) = total outcomes
permutataion
an arrangement of items in which the order of the objects is important
factorial
the result of multiplying a sequence of descending natural numbers; symbolized as n! for any number n
permutation notation
the number of permutations of n items of a set arranged r items at a time is the quotient of n factorial and the factorial of the difference of n and r
combination
a selection of items in which order is not important
combination notation
the number of combinations of n items chosen r at a time is the quotient of n factorial and the product of r factorial and the factorial of the difference of n and r
compound event
an event consisting of two or more simple events (choosing a red sock and a blue sock)
independent events
when one event does not effect the other
dependent events
when one events effects the outcome of another event (without replacement)
probability of A and B
if the A and B are independent events, then the probability of (A and B) is the product of the probability of A and the probability of B
mutually exclusive events
events that cannot happen at the same time
probability of mutually exclusive events
if A and B are mutually exclusive events, then the probability of A and B is zero and the probability of (A or B) is the sum of the probability of A and the probability of B
overlapping events
events with common outcomes
probability of overlapping events
if A and B are overlapping events, then the probability of (A or B) is the difference of (the sum of the probability of A and the probability of B) and the probability of (A and B)
two-way frequency table
displays the frequencies of data in two different categories; also known as a contingency table
conditional probability
probability that an event will occur, given that another event has already occured
expected value
the sum of each outcome's value multiplied by its probability
Calculating Expected Value
If A is an event that includes outcomes A₁, A₂, A₃... and Value(A(n)) is a quantitative value associated with each outcome; the expected value of A is the sum of the products of each outcomes probability and expected value
Experiment
an organized procedure for testing a hypothesis.
sample space
is the set of all possible outcomes
Event
a single outcome or a group of outcomes
Simple Event
one outcome or a single collection of outcomes
Discrete Sample Space
A sample space which is either finite, or countably infinite.
A ∪ B
The set of outcomes in the event A, or B, or both.
A ∩ B
The set of outcomes in the event A and B.
A^c
The set of outcomes that are not in A (A's complement).
The empty event (an event that cannot happen).
mutually exclusive
terms do not overlap
P(S) = 1
P(S) - Probability of the Sample Space
If A ⊂ B...
P(B \ A) = P(B) - P(A)
P(A) - Probability of an Event A
P(A) = P(A₁) + P(A₂) + P(A₃)..., where A₋n is an elementary subset of the event A.
|a|
The number of outcomes in an event A.
List the Sample Space Method
Find the probability of an event by listing the sample space, then dividing the number of elements in the event by the total number of elements in the samples space.
Basic Principle of Counting
If operation 1 can be done in m ways, and operation 2 can be done in n ways, then the combined operation can be done in m * n ways.
Permutations
The number of ordered arrangements of n distinct objects, taken r at a time.
Combinations
The number of unordered combinations of n objects, taken r at a time.
Partitioning Objects into Distinct Groups (where each group is of a given size)
To partition n objects into k distinct groups, with each group containing n₋i objects:
P(A|B) = P(A ∩ B) / P(B)
P(A|B) - Probability of Event B given that an Event A has occurred
P(B|B)
P(B|B) = 1
P(A^c|B)
P(A^c|B) = 1 - P(A|B)
P(C ∪ D|B)
P(C ∪ D|B) = P(C|B) + P(D|B)
Partitions
Multiple events form a partition of the sample space if they are pairwise mutually exclusive, and the union of the events is the entire sample space.
Law of Total Probability
P(A) = ∑(i=1 to n) P(A|B₋i)P(B₋i)
Bayes Rule
The probability of a subset B₋k of a partition, given an event A:
Random Variable
If X is a random variable, then P(X = x) is the probability of seeing a specific value in the sample space.
Discrete Random Variable
A random variable where the range of values that the variable can take on is a countable set
Probability Function
f(x) = P[X = x], x ∈ Rx (the range of values x can take on)
E(x)
Expected Value of a DRV
Var(x)
Variance of a DRV
Bernoulli Random Variable
A random variable that can only take on one of two values.
Trial
each result/observation of an experiment, such as one roll of a number cube.
Binomial Distribution
Used to find the probability of x successes in n trials.
Geometric Distribution
Used to find the probability that the first success happens on the xth trial.
Negative Binomial Distribution
Used to find the probability that the rth success happens on the xth trial.
Hypergeometric Distribution
Used to find the probability of choosing exactly x elements of a given type r out of a population of N elements to fill a subset of n elements.