Probability
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Probability
The subject of probability is an integral part of all work with statistics. Research in all fields requires that we use inferential statistics. (We need to know how to use data from a sample to infer characteristics of a population.) But even the best selected sample can be ill-reflective of the population from which it was selected for a number of reasons. Therefore, when making conclusions about the data of a sample, we would also like to attach a certain level of likelihood that the sample reflects the population as accurately as possible.
We need probability to do this. If there was a 20% chance that we could be wrong, we’d likely go back and redraw the sample. But, if there was a 1% chance we could be wrong, we'd likely go with our findings.
Definitions:
Probability: - A number which can be expressed as a percent or decimal. It must be between 0 and 1. Ex. If there is a 20% chance of rain, this is expressed as .20, which is between 0 and 1.
Random Experiment: An observational process whose results cannot be known in advance.
Outcome: Refers to the eventual outcome of a single experiment
Sample Space: The set of all probable outcomes of a given experiment
Event(s): Refers to any subset of outcomes in the same space.
Simple Event: A single outcome
Compound Event: consists of two or more simple events
Notation for Probability:
S: The Sample Space
P: Probability
Capital letter following P: usually given to denote a specific event. Ex. The probability (P) of getting our tests back (T) tomorrow lies between 0 (impossible) and 1 (certain):
0 ≤ P(T) ≤ 1
Venn Diagram: A method of illustrating relationships between sets of units. Usually drawn with circles representing the boundary of a set.
A B
ᴜ = Union of two sets. In a Venn Diagram, this notation is read as "or". Ex. The unit is in set A or set B.
∩ = The intersection of two sets. In a Venn Diagram, this notation is read as "and". Ex. The unit is in set A and set B
f = (meaning frequency), the number of ways the event can occur
n = total number of outcomes
TOPICS
5.1 Random Experiments
A random experiment is an observational process whose results cannot be known in advance. Ex. Tossing a single die is a random experiment. The sample space is the set of all possible outcomes for the experiment. In the example of tossing a single die, the sample space would be expressed as:
S = {1, 2, 3, 4, 5, 6}
Because there are only 6 sides of a die, each bearing a numerical representation between 1 and 6.
The units in an experiment may be discrete, as in the sides of a die, (whole numbers) or they may be continuous, like a collection of GPAs (fractional numbers which, by definition, are never ending). If the outcome of an experiment is a continuous measurement, the sample space cannot be listed. (too large). It can only be described by rule.
S = {all X such that X ≥ 1)
For example, the sample space to describe a randomly chosen student's GPA would be:
S = (all X such that 0.00 ≤ X ≤ 4.00)
Event: Any subset of outcomes in the sample space. A simple event is a single outcome. Ex. Tossing a coin. The sample space is S = {heads, tails}. We say that these two simple events are equally likely. However, when buying a lottery ticket, the two simple events in that sample space of {win, lose} are not equally likely.
Simple events are the building blocks on which we can define a compound event consisting of two or more simple events. For ex. Amazon's "Books & Music" has seven categories that a shopper might choose: S = {Books, DVD, VHS, Magazines, Newspapers, Music, Textbooks}. Within this sample space, we could define compound events, such as "electronic media" as A = {Music, DVD, VHS} and "print periodicals" as B = {Newspapers, Magazines}.
This can be shown in a Venn Diagram.
5.2 Probability
The probability of an event is a number that measures the likelihood that the event will occur. The probability of event A, denoted P(A), must lie within the interval of 0 (impossible) and 1 (certain).
There are three distinct ways of attaining probability.
Approach
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How Assigned
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Example
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Empirical
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Estimated from observed outcome frequency
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There is a 2 percent chance of twins in a randomly chosen birth.
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Classical
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Known a priori by the nature of the experiment
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There is a 50 percent chance of heads in a coin flip
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Subjective
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Based on informed opinion or judgment
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There is a 75 percent chance that England will adopt the Euro currency by 2012
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Empirical Approach:
If we have collected the data through observation or experiment. This is also called the "relative frequency" approach. (If you will recall, relative frequency is expressed as a percent and obtained by dividing the number of items in the bin by the total number of items in the sample.) We conduct the empirical approach the same way, dividing the frequency of observed outcomes (f) by the total number of observations (n). Or, f/n
Ex. We could estimate the reliability of a bar code scanner:
P(a missed scan) = number of missed scans/number of items scanned
As we increase the number of times we perform the experiment/trials (n), our estimate will become more accurate.
The Law of Large Numbers: An important probability theorem which says that as the number of trials increases, any empirical probability approaches its theoretical limit. Ex. Coin flip. Flipping a coin should produce heads .50 of the trials. However, in a small number of trials, it's not likely to be the case. (More like 7 out of 13 or 28 out of 60). A very large number of n may be needed before f approaches .50
Classical Approach: a priori - "from before". This approach assigns probability before any experiment is carried out because of past results. Ex. Flipping a coin or rolling a die allows us to envision the entire sample space and therefore deduce probability of any particular outcome.
Ex. Throwing two dice has 36 possible outcomes, so the probability of rolling a seven is known to be:
P(rolling a seven) = number of possible outcomes with 7 dots/number of outcomes in sample space = 6/36 = 0.1667
Subjective Approach: Reflects someone's informed judgment about the likelihood of an event. It is needed when there is no repeatable random experiment. For ex. What is the probability that Ford's new supplier of plastic fasteners will be able to meet the Sept. 23 shipment deadline? In such a case, we rely on personal judgment or expert opinion. That judgment is likely based on experience with similar events and knowledge of the underlying causal process.
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