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AHSS Ch. 3.1-3.3 Probability
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Key Concepts:
Terms in this set (27)
Relative Frequency Probability
The proportion of times the outcome would occur if we observed the random process an infinite number of times.
Relative Frequency
The proportion of times the event occurs out of
the number of trials.
Law of Large Numbers
As more observations are collected, the observed proportion of occurrences with a particular outcome after n trials converges to the true probability P of that outcome.
Disjoint or Mutually Exclusive
The relationship of two outcomes that cannot both happen in the same trial, i.e. P(A and B) = 0.
Addition Rule for disjoint events
If outcomes are disjoint, the probability that one or the other occurs is found by adding their individual probabilities.
Venn Diagrams
Useful when outcomes can be categorized as "in" or "out" for two or three variables, attributes, or random processes.
General Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
∩
Symbol for intersection / "and"
U
Symbol for union / "or" (inclusive)
Sample Space (S)
The set of all possible outcomes. The total probability of a sample space must equal 1.
Complement
An event A together with its complement comprise the entire sample space.
Independent events
The outcome of one does not change the likelihood of the outcome of the other, i.e. P(A | B) = P(A).
Multiplication Rule for independent events
The probability of 2
independent
events can be calculated as the product of their unconditional probabilities.
Conditional Probability
The probability of an event is computed under another condition (a known outcome or event). The outcome of interest A given condition B is
computed as:
P(A|B) =P(A and B)/P(B)
Joint Probability
A probability that measures the likelihood two or more events will happen concurrently.
Joint/And Probabilities from a tree diagram
When there are two events, this is found as P(A) x P(B | A)
General Multiplication Rule
P(A and B) = P(A)
P(B|A) = P(A|B)
P (B)
Independent Events Check
If one of the following holds true: P(A|B)=P(A) ,
P(A and B)=P(A)*P(B), then A and B are independent
Mutually Exclusive (Disjoint) Check
If one of the following holds true: P(A and B) = 0,
P(A or B) = P(A) + P(B), then A and B are mutually exclusive (disjoint)
Tree Diagrams
A branched diagram used to organize outcomes and probabilities; most useful when two or more processes occur in a sequence and each process is conditioned on its predecessors.
Bayes' Theorem
A theorem used to find inverse probabilities. It can be derived using a tree diagram (see diagram)
• The numerator identifies the probability of getting both A and B.
• The denominator is the overall probability of getting B. Traverse each branch of the tree diagram that ends with event B. Add up the required products.
special case of conditional probability: P(A|B) =P(A and B)/P(B)
Binomial Formula Check
Four conditions to check.
(1) The trials are independent.
(2) The number of trials, n, is fixed.
(3) Each trial outcome can be classified as a success or failure.
(4) The probability of a success, p, is the same for each trial.
Binomial Formula
Suppose the probability of a single trial being a success is p. Then the probability
of observing exactly x successes in n independent trials is given by
Binomial coefficient
n choose x = n!/[x!(n-x)!]; computes the number of combinations with exactly x success in n trials:
primary branch of tree diagram
Corresponds to unconditional probabilities of the form P(A)
secondary branch of tree diagram
corresponds to conditional probabilities of the form P(B | A)
P(at least 1)
Solve via complements; P(at least 1) = 1 - P(none)
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