Chebyshev and Central Limit Theorem

**morganfor** · August 2nd 2009, 08:08 PM

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!!!

**matheagle** · August 2nd 2009, 08:17 PM

n=1,000,000 and p=.5 approximating this binomial with a normal (CLT) you need

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

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

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

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

**morganfor** · August 4th 2009, 03:12 PM

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?

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