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Math Help - fair coin probability

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    fair coin probability

    What is the probability that in 120 tosses of a fair coin between 40% and 60% (inclusively) will be heads?
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    Quote Originally Posted by swoopesjr01
    What is the probability that in 120 tosses of a fair coin between 40% and 60% (inclusively) will be heads?
    Note,
    40% of 120 is 48
    60% of 120 is 72
    Which means the probability is,
    P(48)+P(49)+...+P(71)+P(72)=\sum^{72}_{k=48}P(k)
    But, because of binomial probability we have,
    P(k)={120 \choose k}(1/2)^k(1/2)^{120-k}={120 \choose k}(1/2)^{120}
    Thus,
    \sum^{72}_{k=48}{120 \choose k}(1/2)^{120}\approx .9779
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    Quote Originally Posted by swoopesjr01
    What is the probability that in 120 tosses of a fair coin between 40% and 60% (inclusively) will be heads?
    Use the normal approximation to the Binomial distribution for this.

    The mean number of heads is 60, the standard deviation is
    \sqrt{120\times 0.5\times 0.5}=\sqrt{30}

    Between 40% and 60% of 120 is between 48 and 72. To allow for the
    fact that we can only have an integer number of heads we work the
    normal approximation (which is continuous) for between 47.5 and 72,5
    heads.

    The z-scores for the ends of the range are:

    <br />
z_{lo}=\frac{47.5-60}{\sqrt{30}}\approx -2.282<br />

    <br />
z_{hi}=\frac{72.5-60}{\sqrt{30}}\approx 2.282<br />

    Now looking the probability of a standard normal random variable having
    a value in the range [-2.28,2.28] we find the probability is
    \approx 0.9775.

    This last answer may be compared with PH's calculation based on using
    the exact binomial formula.

    RonL
    Last edited by CaptainBlack; May 23rd 2006 at 08:41 PM.
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    Man, I really wish I learned the normal curve theory I look like a noob when I answer these probability questions.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ThePerfectHacker
    Man, I really wish I learned the normal curve theory I look like a noob when I answer these probability questions.
    Pffl. I don't know nothin' "normal!"

    -Dan
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