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    I've managed to confuse myself...How do we know that the probability of getting a head after one toss of a coin is 0.5? Is it only via experiment?

    You could predict which way up a coin will land if you knew all the variables (assume a machine tosses it so we can discard the "free will" element). So to talk of the probability of getting a head seems only to make sense if you can't predict the outcome beforehand.

    So probability just models situations you can't explain from the outcomes of previous similar situations, and the probability of an event changes depending on what you know?
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    because there are only two options, the coin has to land one way or the other, and it defiantly will fall, so;
    p(head) + p(tail) has to = 1

    Coins are balanced so there is no reason for one side to always land face down, so we say the coin is fair, this also means that the total probability of 1 is shared equally between the two outcomes, hence 1/2 = 0.5
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    (Original post by jamie092)
    You could predict which way up a coin will land if you knew all the variables (assume a machine tosses it so we can discard the "free will" element). So to talk of the probability of getting a head seems only to make sense if you can't predict the outcome beforehand.
    So probability just models situations you can't explain from the outcomes of previous similar situations, and the probability of an event changes depending on what you know?
    Essentially, yes. We assume for most purposes that we don't have enough information to change our prediction of the coin appreciably (if I flipped a coin right now, I probably am in possession of enough information to weight the probability by about 0.00001% or something like that, if I did loads of calculations). If I know that I will throw a head (perhaps I know that there's a magnet on the top side of the coin and a corresponding magnet of opposite polarity on the bottom, and a further magnet on the floor), then I can still estimate the probability that I will throw a head (as 1) - the probability still exists, it's just not very interesting. By which I mean: it makes mathematical sense to predict probabilities even when you can predict the outcome beforehand; it just doesn't help.
    Re your last paragraph: the probability changes depending on what you know, in a precise way which is formulated by Bayes' Law.
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    (Original post by jamie092)
    I've managed to confuse myself...How do we know that the probability of getting a head after one toss of a coin is 0.5? Is it only via experiment?
    If you have n equally likely outcomes, then the probability of a chosen outcome is \frac{1}{n}. (This is because all the probabilities must be equal, and the total of all the probabilities must be 1.)

    For the coin, we have two equally likely outcomes, so the probability of getting a chosen one of them is \frac{1}{2} = 0.5.

    The assumption that you're questioning is that heads and tails are equally likely. Since we haven't got any way of predicting what is going to happen. and there's nothing special about either outcome that would make it more likely than the other, the assumption is fair enough. If you want to get philosophical about it, then we could just say that the probabilities reflect our lack of knowledge about the system. (Incidentally, this is no longer true in quantum mechanics, where events occur that are genuinely probabilistic!)
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    Oh ok so given no other information we assume the probabilities are equal. That makes sense.

    I feel stupid now cause I've done a module on Probability and Statistics which covered Bayes' theorem and I did well in the exam but I didn't properly learn much of it apart from how to apply the formula.

    I've heard about experiments that prove "true randomness" in quantum mechanics but I haven't actually studied it enough to have a proper opinion on it. From experience of everything else I've come across though, I have a gut feeling that they'll find a cause for their results though.
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    (Original post by jamie092)
    I feel stupid now cause I've done a module on Probability and Statistics which covered Bayes' theorem and I did well in the exam but I didn't properly learn much of it apart from how to apply the formula.
    Bayes' Theorem is hard to grasp - the best explanation I've ever read of it that tries to make it intuitive is http://yudkowsky.net/rational/bayes
    It's long; the author describes it as "excruciatingly gentle" but he is ridiculously intelligent, so it may take a while to grok properly. I'd recommend reading the first couple of paragraphs, and if you get bored, skip the rest and make sure you can do the exercises on the page; when you've done the exercises, read his explanations. That *really* helped me - before, I knew BT as an abstract theorem, but now I half-grok it.
 
 
 
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