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What is Probability?. American Heritage Dictionary Definition 3: “ Math . A number expressing the likelihood of occurrence of a specific event, such as the ratio of the number of experimental results that would produce the event to the total number of results considered possible.”

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what is probability
What is Probability?
  • American Heritage Dictionary Definition 3: “Math. A number expressing the likelihood of occurrence of a specific event, such as the ratio of the number of experimental results that would produce the event to the total number of results considered possible.”
  • AHD Definition 1 of Likelihood: “The state of being likely or probable; probability.”
from the web
From the web
  • “Probability is the name given to the branch of mathematics that deals with chance and how to predict whether a result is likely or unlikely.”(http://www.learn.co.uk/default.asp?WCI=Unit&WCU=275)
  • “By probability, we generally mean the likelihood of a particular event occurring, given a particular set of circumstances.The probability of an event is generally expressed as a quantitative measurement.”(http://www.maps.jcu.edu.au/hist/stats/quet/quet6.htm)
compare
Compare:
  • What is time?
  • What is a point?
probability of an event three perspectives
Probability of an Event: Three Perspectives
  • Classical (“A priori” or “theoretical”)
  • Empirical (“A posteriori” or “Frequentist”)
  • Subjective
classical probability a priori or theoretical
Classical Probability (“A Priori” or “Theoretical”)
  • Situation: “experiment” or “random process” with n equally likely outcomes.
  • E.g, toss a fair die: Six equally likely outcomes,
  • P(A) = m/n, where A is satisfied by exactly m of the n outcomes
  • E.g., toss a fair die; A = an odd number comes up -> P(A) = 3/6.
pros and cons of classical probability
Pros and Cons of Classical Probability
  • Conceptually simple for many situations
  • Doesn’t apply when outcomes are not equally likely.
  • Doesn’t apply when there are infinitely many outcomes
empirical probability a posteriori or frequentist
Empirical Probability (“A Posteriori” or “Frequentist”)
  • P(A) = limn --> ∞(m/n), where n = number of times process performed, m = number of times A is satisfied.
  • E.g., toss a fair die; A = six lands up
  • E.g., toss a die that is suspected of not being fair; A = six lands up.
pros and cons of empirical probability
Pros and Cons of Empirical Probability
  • Covers more cases than classical.
  • Intuitively agrees with classical when classical applies.
  • Repeating the identical experiment an infinite number of times (sometimes even twice) is physically impossible.
  • How large must n be to give a good approximation to the limit?
subjective probability
Subjective Probability
  • A person’s measure of belief that some given event will occur.
  • E.g., P(the stock market will go up tomorrow).
  • Needs to be “coherent” to be workable. (e.g., P(stock market goes up tomorrow) = .6 and P(stock market goes down tomorrow) = .7 are inconsistent.)
pros and cons of subjective probability
Pros and Cons of Subjective Probability
  • Applicable in situations where other definitions are not.
  • Fits intuitive sense of probability.
  • Can be considered to extend classical.
  • Can vary from individual to individual
  • Requires “coherence” conditions; are people always that rational?
unifying perspective axiomatic model of probability
Unifying Perspective: Axiomatic Model of Probability

A function P from events to non-negative numbers satisfying:

  • 0 ≤ P(E) ≤ 1
  • P(S) = 1 (S = certain event; sample space)
  • P(union of mutually exclusive events) = sum of P of individual events
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