Let S be the number of heads in 1,000,000 tosses of a fair coin. How does on use (a) Chebyshev's inequality and (b) the Central Limit Theorem to estimate the probability that S lies between 499,500 and 500,500?

Thanks so much!!!

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- Aug 2nd 2009, 08:08 PMmorganforChebyshev and Central Limit Theorem
Let S be the number of heads in 1,000,000 tosses of a fair coin. How does on use (a) Chebyshev's inequality and (b) the Central Limit Theorem to estimate the probability that S lies between 499,500 and 500,500?

Thanks so much!!! - Aug 2nd 2009, 08:17 PMmatheagle
n=1,000,000 and p=.5 approximating this binomial with a normal (CLT) you need

$\displaystyle \mu=np$ and $\displaystyle \sigma^2=npq$

Since you want the probability of an event that has $\displaystyle \mu$ at its center this is straightforward.

$\displaystyle P(499,500 < X_B < 500,500)\approx P\biggl({499,500-\mu\over\sigma} < Z < {500,500-\mu\over\sigma}\biggr)$

As for cheby's figure out your k at http://en.wikipedia.org/wiki/Chebyshev's_inequality

look under Probabilistic statement - Aug 4th 2009, 03:12 PMmorganfor
I'm just confused because for this problem doesn't k=1 for Chebyshev so that 1/k^2 = 1 which means that there is a 0 probability that S lies between 499,500 and 500,500?