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| the change or likelihood that something is the case or will happen |
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| process of observation or measurement |
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the result of an experiment
*if the outcome of an event is not certain, we can discuss process of observation in which the probabilities of particular outcomes are considered |
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the list of all possible outcomes of an experiment
1. categories do not overlap
2. no result is classified more than once
3. the list is complete |
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- a visual representation of the experiment
- used to list all possible outcomes |
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| how likely it is that an outcome in a sample space will occur |
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| the probability that event A will occur |
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| properties of probability |
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1. probabilities are always between 0 and 1, inclusive
2. the sum of all the probabilities in a sample space is 1 |
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| an event with a probability of 0 |
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| an event with a probability of one |
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| theoretical probabilities |
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| probabilities we assign to outcomes based on our understanding of the sample space |
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- each possible outcome is equally likely to occur
in general, s= {A1, A2, ..., Am} where there are m outcomes.
P(S)= P(A1) + P(A2) + ... + P(Am)
- if outcomes are equally likely
P(s)- m x p=1 m x p=1 so p= 1/m |
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Definition
in a uniform sample space with m outcomes, each outcome has a probability of 1/m.
P(A)= 1/n(s)=1/m
where n(s) is the number in the sample space |
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- "experimental probability"
- use when theoretical probability is not possible or as an alternative to theoretical probability
*as N increases, the empirical probability gets closer and closer to the theoretical probability |
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| empirical probability formula |
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P(A)=number of times A occurs/N
n= number of trials; should be very large |
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| a subset of the sample space |
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| if S is a uniform same space |
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n(s)= number of equally likely outcomes
n(a)= number of outcomes in event A
P(A)= n (A)/ n (s) |
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A or B
A U B
in A or B or both |
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| the collection of all outcomes in the sample space that are not in A |
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| mutually exclusive events (disjoint sets) |
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1. events A and B are mutually exclusive if they have no outcomes in common
2. if events A and B are mutually exclusive
P (A U B)= P(A) + P(B)- (P A and B) |
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| probability of the complement of A |
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P(A I B)
The probability of even A given that event B has occurred |
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| conditional probability formula |
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| P(A I B)= P(A and B) / P(B) |
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| If the odds favoring an event E are m to n then, |
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P(E)= m / m + n
P(not E)= n / m + n |
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| performed as a sequence at consecutive steps |
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| the average payoff per experiment when the experiment is performed a large number of times (the expected value may actually be unlikely or impossible). |
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number of ordered subsets of a set
order matters! |
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| number of permutations of n objects taken r objects at a time |
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| number of unordered subsets of a set |
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| the number of combinations of n objects taken r objects at a time (formula) |
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