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the mass distribution of a coin is so that the chance of getting a head s only 40% .The coin is tossed a 100 times what is the chance -to get more than (inclusive ) 50 heads ? -to get more than (exclusive) 50 tails? to get more than (exclusive) 60 tails and more than (exclusive ) 20 heads ?

if you do the experiment 3times,what is the chance of getting every time less than (inclusive ) 30 heads ?

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    This looks like homework. Please see http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question and update your question accordingly. Also, it would be nice if you checked your question for spelling, grammar, punctuation and capitalization.2011-09-21

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To get you started, I will work the first one.

Using the Binomial Distribution, we get $$ \sum_{k=50}^{100}\binom{100}{k}\;.4^k\;.6^{100-k} = 0.027099197757009005051 $$ If we approximate, using the Normal Distribution, where the mean is $-.2\times100=-20$ and the variance is $.96\times100=96$, the probability of getting at least $0$ ($20/\sqrt{96}$ s.d. above the mean) is $\frac{1}{2}(1-\operatorname{erf}(20/\sqrt{96}/\sqrt{2}))=0.020613416668581847098$.

As Brian Scott points out in the comments, we can get a better approximation with the Normal Distribution by including the whole count around $50$ rather than only half the count. That is, the probability of $49.5$ heads or more. $49.5\text{ heads}-50.5\text{ tails}=-1$ instead of $0$, which gives $19/\sqrt{96}$ or more s.d. above the mean, with a probability of $\frac{1}{2}(1-\operatorname{erf}(19/\sqrt{96}/\sqrt{2}))=0.026239749779623305681$.

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    You can do a better normal approximation. For one toss $\mu = 0.4, \sigma \cong \sqrt{0.4\cdot 0.6} = \sqrt{0.24}$; for $100$ tosses $\mu = 40, \sigma \cong 10\sqrt{0.24}$. You need to be about $9.5/\sigma\cong 1.94$ s.d. above the mean $-$ $9.5$ to account for including $50$ $-$ so from [this table](http://itl.nist.gov/div898/handbook/eda/section3/eda3671.htm) the probability is about $0.5 - 0.0.47381 = 0.02619$.2011-09-21
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    @Brian: I think we are using different Normal Distribution models. I am putting a head at $+1$ and a tail at $-1$. Then the mean comes to $.4(+1)+.6(-1)=-.2$ and the variance to $(.4(+1)^2+.6(-1)^2)-(-.2)^2=.96$. $50$ heads and $50$ tails is $0$, which is $20$ above the mean, but I guess I could have added some discrete slop by considering more than $49.5$ heads (and $50.5$ tails) which would give a score of -1 or only $19$ above the mean: $\frac{1}{2}(1-\operatorname{erf}(19/\sqrt{96}/\sqrt{2}))=0.026239749779623305681$. This is closer.2011-09-21
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    I understood that you were looking at #(heads $-$ tails); I was commenting on the lack of discrete slop and was too lazy to rearrange the calculation into your format.2011-09-21
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    @Brian: I figured you knew. I was simply taking the opportunity to explain (mostly to the OP) what I had done. I hadn't really explained how I was counting heads and tails in my original answer. I had thought about the "discrete slop" a bit after I had originally posted, but I had to go AFK before I could correct either omission.2011-09-22
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    It seems quite unnatural to use erf instead of $\Phi$ in this context. At any rate, one should do a continuity correction, i.e. $\ge 50$ is the same as $>49$ in this context, so with the continuous approximation, use $49.5$. This gives surprisingly accurate approximations even for samples as small as 6, and in fact it's astonishing how accurate it is when the sample size is just $1$. If you get $1$ with probability $p$ and $0$ with probability $1-p$, then the expected value is $p$ and the standard deviation is $\sqrt{p(1-p)}$. So the probability that a normally distributed ("Gaussian")...2011-09-22
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    ...random variable with that expected value and that standard deviation is more than $0.5$ should approximate $p$. Here's the surprise: it never differs from $p$ by more than about $0.03$.2011-09-22
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    @Michael: erf seems to be the standard function. I use $\frac{1}{2}(1+\operatorname{erf}(x/\sqrt{2}))$ for the Cumulative Normal Distribution, since I was unaware that $\Phi$ was a standard notation for it (at least as standard as $\operatorname{erf}$?). However, I did use $49.5$ in the latter part and tried to justify why it was better than $50$ in this context. I called it "discrete slop".2011-09-22
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All right, I'll try to write this a bit more neatly than in robjohn's answer. He fully disposed of the exact answer; I'll address the approximation by the normal distribution.

It was Abraham de Moivre in the first half of the 18th century who introduced the bell-shaped curve into probability theory, showing that the density is $\varphi(x) = \text{constant}\cdot e^{-x^2/2}$ and finding the constant numerically. His friend James Stirling showed that the constant is $1/\sqrt{2\pi}.$ Today, the lower-case $\phi$ is standard notation for this function and capital $\Phi$ is its antiderivative that approaches $0$ at $-\infty$ and $1$ at $\infty$.

When one tosses a biased coin, one gets 1 head with probability $p$, or 0 heads with probability $1-p$. The expected value of that probability distribution is $0\cdot(1-p)+1\cdot p = p$. The standard deviation is $\sqrt{p(1-p)}$, and I leave the details to you unless you ask.

When one tosses it 100 times, and the tosses are independent, the expected value is $100p$ and the standard deviation is $\sqrt{100p(1-p)}$. For $p=0.4$, the expected value is 40 and the standard deviation is $\sqrt{100p(1-p)}\approx 4.898979485566$.

The probability that this is at least 50 is also the probability that it is strictly more than 49. The "continuity correction" calls for using the number half-way between those when doing the continuous approximation. That is $49.5$.

So the probability of getting 49 or fewer successes is $$ \Phi\left(\frac{49.5 - 40}{4.898979485566}\right) \approx 0.97376. $$ The probability of getting 50 or more successes is 1 minus that number.

The exact answer for $\le49$, if I can believe the software I'm using is about $0.9729008$.

If you use the continuity correction, then even with a sample of size 1 (where it would be silly---and worse, it would be pointless---to use the continuous approximation) the continuous approximation never differs from the exact result by more than about $0.03$. With six coin tosses, it gives somewhat respectable results. With 100, it is reasonably accurate.

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    "...James Stirling showed that the constant is $1/\sqrt{2\pi}$" - Are you referring to the constant $\sqrt{2\pi}$ in the Stirling approximation or the normalizing constant of the Gaussian distribution?2011-09-23
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    Yes. They're practically the same thing. You can get de Moivre's result from Stirling's formula and vice versa. In fact, I think de Moivre may be the original discoverer of Stirling's formula, except for a closed-form expression for the constant.2011-09-23