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Thread: Probabilties and Infinities

  1. #1 Probabilties and Infinities 
    Forum Freshman CrimsonViper's Avatar
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    Ok bear with me on this one as I haven't studied Maths at an Academic Level for two years (i was 16 when I gave it up).
    I am currently reading a book called "The Infinite Book" by John D. Barrow and is agood book which I recommend to anyone interested in Popular Science books.
    It got thinking. If I flip a coin, the probability of it landing on heads is 1/2 (or 0.5). If I flip two coins the probablity of getting 2 heads would be 1/4 (or 0.25).
    Now lets say I am immortal so have a boring eternity on my hands and i decide to spend my time tossing the coin an infinite amount of times. By some sheer fluke, every single toss lands on heads. The chances of this happening are 1/infinite (or 0) so my sheer fluke has beaten a Mathematical impossiblity. Has my Maths gone wrong somewhere or have I stumbled across a paradox of an infinity? More likely it is the former.


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  3. #2  
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    you can not flip a coin an infinite amount of times. chances of you flipping a heads every flip is exactly equal to where n is the number of times you flip the coin. the issue comes with the fact that infinity is not a number, nor a quantity.


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  4. #3  
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    There are other cases where a similar paradox appears. For example choose a number at random between 0 and 1. The probability for any specific number is zero, but some number does get chosen.
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  5. #4  
    Forum Radioactive Isotope MagiMaster's Avatar
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    That's only a paradox in the sense of being counter-intuitive, not in the sense of there being any contradiction.
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  6. #5 Re: Probabilties and Infinities 
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    Quote Originally Posted by CrimsonViper
    Ok bear with me on this one as I haven't studied Maths at an Academic Level for two years (i was 16 when I gave it up).
    I am currently reading a book called "The Infinite Book" by John D. Barrow and is agood book which I recommend to anyone interested in Popular Science books.
    It got thinking. If I flip a coin, the probability of it landing on heads is 1/2 (or 0.5). If I flip two coins the probablity of getting 2 heads would be 1/4 (or 0.25).
    Now lets say I am immortal so have a boring eternity on my hands and i decide to spend my time tossing the coin an infinite amount of times. By some sheer fluke, every single toss lands on heads. The chances of this happening are 1/infinite (or 0) so my sheer fluke has beaten a Mathematical impossiblity. Has my Maths gone wrong somewhere or have I stumbled across a paradox of an infinity? More likely it is the former.
    Probablility 0, despite common terminology, does not mean impossible.

    Here is why.

    In a general way, the probability of some event happening is the ratio of the number of times that it happens to the number of independent trials, in the limit as the number of trials tends to infinity. So if that limit is zero, then the probability of the event is zero, even though it may occur, and actually occur infinitely many tmes, in that hypothetical, infinite number of trials.

    It is easy to see that the probability is 0 if the event occurs only finitely many times. If you spread the number of trials out a bit, then with a little thought you can see how infinitely many occurrences might happen in an iinfinite number of trials. For instance if the event occurs at each trial then the probability will be zero since tends to 0 as becomes large.
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  7. #6 Re: Probabilties and Infinities 
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    QUESTION
    Assuming you have six marbles inside a bag (4 red and 2 blue); you select one red marble at random, what will be the probability without replacing it?
    What is the probability of selecting a blue marble without replacing it?

    ANSWER
    P(red)= 3/5

    P(blue)= 1/5

    How can this answer be explained to learners?

    Comment Posted by Bafedile
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  8. #7 Re: Probabilties and Infinities 
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    Quote Originally Posted by Bafedile
    QUESTION
    Assuming you have six marbles inside a bag (4 red and 2 blue); you select one red marble at random, what will be the probability without replacing it?
    What is the probability of selecting a blue marble without replacing it?

    ANSWER
    P(red)= 3/5

    P(blue)= 1/5

    How can this answer be explained to learners?

    Comment Posted by Bafedile
    You could start by stating the questions more clearly.

    For the first question you ask "what will be the probability", without saying of what.

    For the second question, the sequence of steps leading up to the blue marble is unstated.
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  9. #8 Re: Probabilties and Infinities 
    . DrRocket's Avatar
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    Quote Originally Posted by mathman
    Quote Originally Posted by Bafedile
    QUESTION
    Assuming you have six marbles inside a bag (4 red and 2 blue); you select one red marble at random, what will be the probability without replacing it?
    What is the probability of selecting a blue marble without replacing it?

    ANSWER
    P(red)= 3/5

    P(blue)= 1/5

    How can this answer be explained to learners?

    Comment Posted by Bafedile
    You could start by stating the questions more clearly.

    For the first question you ask "what will be the probability", without saying of what.

    For the second question, the sequence of steps leading up to the blue marble is unstated.
    I think he is simply removing one red ball and then asking what is the probability of picking a red or blue ball. In that case his answer is wrong and the probabilities are

    P(red)= 3/5

    P(blue)= 2/5

    This should be patently obvoius since what you have is a bag with 3 red balls and 2 blue balls.

    It is a good sign if the sum of the probabilities mutually exclusive and exhaustive events is 1 and a bad sign otherwise.
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