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!!!
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!!!
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