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Probability Axioms

  1. Jul 31, 2013 #1
    Axioms are: -
    1) P(E) >= 0
    2) P(S) = 1
    3) P(E1 U E2 U ...) = P(E1) + P(E2) + .... if all are mutually exclusive

    Why are the axioms defined in such a way? Why not this simple axiom: - Probability of an event is number of favorable outcomes divided by total number of outcomes?
  2. jcsd
  3. Jul 31, 2013 #2


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    That would assume that all events are equally likely, which - in general - they are not.
    If S = { today it rains, today it doesn't rain } then P(today it rains) is not 1 / |S| = 1/2. If that were true, replace " it rains" by "we all die in a meteor impact".
    So what you do is assign a probability P(s) to every ##s \in S##. The axioms make sure that it matches our intuition.
  4. Jul 31, 2013 #3
    I am still not comfortable with the 3) axiom. It seems a bit indirect to me.
    Suppose we toss a coin and we want to find the probabilities of heads and tails. Now P(H) + P(T) = 1 ... from 3)
    Since both are equally probable both are equal and hence P(H) = P(T) = 1/2
    It is all indirect. We could have directly said that out of two possibilities head or tail is one and thus it is 1/2
  5. Jul 31, 2013 #4
    Suppose you have a coin which will be tossed (so [itex]S=\{\text{heads}, \text{tails}\}[/itex]), and it's weighted so that the probability of heads is 52%.

    Q1) Does this seem like a plausible situation?
    Q2) Does it seem plausible that mathematics can inform ones decisions of which bets to take concerning this coin?
    Q3) What do you think is a reasonable answer to: "What's the probability of tails?"
  6. Aug 1, 2013 #5
    Yes, although I am a bit unsure what you are asking.

    Sorry but I couldn't find any relevance to my question earlier.
  7. Aug 1, 2013 #6


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    What would the probability of tails be for this rigged coin, using your definition above?
  8. Aug 1, 2013 #7

    D H

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    The relevance is that you used the third axiom to calculate that 48% figure.
  9. Aug 1, 2013 #8
    Thinking more about it I realised the importance of the 3rd axiom. Nice example to make me understand.
    Many Thanks!
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