Probability I - Definitions I

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Probability

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Probability is the mathematical study of chance. The probability of an event is symbolized as Pr(x), where x is the event. Probabilities are measured using a statistical measure called the proportion, symbolized as p. Probabilities are based on the notion of likelihood.

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Chance

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Events for which there is no known cause. Even if it is believed that every event has a cause, often something will happen for no reason that we know of. Sometimes we say that this happened "by chance." When there is no clear line of simple causes, we use the word "chance."

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Odds

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Odds are used to express likelihoods more exactly, with numbers. Odds are based on a statistic called the ratio, which in turn is based on the statistic called the count. Likelihoods cannot always be calculated using counts alone, but when they can be, the notion of odds is referenced. Odds are expressed with the count of the items of interest, n, followed by a colon (:), followed by the remainder, N-n. For example there are 12 total sheep, 5 of which are black and 7 are white. The odds that any one of the sheep we come across will be black are five to seven (5:7). n:N-n Example P = n/N = 5/12 = 0.4167

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Ratio

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Rations express the relationship between two counts. They are exact measures of just how much bigger or smaller one number is than another. Ratios are expressed as numbers in three principal ways, as proportions, as percentages, and as odds.

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Classical Probability

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Relies on the notion that all events are equally likely.

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Frequentist probability

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Relies on the notion of replication. Frequentist probability represents a long-run frequency of an event such as an infinite number of coin tosses resulting (presumably) in the probabiity of a coin coming up heads 0.5 or 50% of the time.

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Subjective Probability

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Relies on the notion of rational choice

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The Rule of Insufficient Reason

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When there is no reason to think that one possibility is more or less likely than another, assume that the two likelihoods are exactly same.

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Traditionally, there are said to be 3 types of Probability. What are they?

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Classical, Frequentist, and Subjective

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Scalability

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The measures of probability must all be between zero and one

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Complements

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The probability of something not happening must be equal to one minus the probability of that same thing happening.

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Addition

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For any group of events, the probability of the whole group must be equal to the sum of the probabilities of each individual event.

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Pr(x)

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The probability of an event.

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Mutually Exclusive Events

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Two events (and the probabilities of those events) are mutually exclusive if the fact that one happens means that the other cannot possibly have happened. This is always true for different values of a single variable. If, for example, the color of a sheep could only be all black or all white, then if the color of the sheep we pick is black, its color cannot be white.

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How are ratios expressed?

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Ratios are expressed as proportions, percentages, and odds.

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Kolmogorov's axioms

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Kolmogorov's axioms consist of the three rules of scalability, complements and addition.