In general what does the chebyshev inequality refers and how can I prove this theorem using integral signs?
as usual I don't understand your questions
what is 'integral signs'?
The proof consists of splitting the integral into two pieces and throwing away one
and using an inequality on the other
http://en.wikipedia.org/wiki/Chebyshev's_inequality
that's a binomial problem that can be fairly approximated with the normal ditsribution via the central limit theorem
n=100, p=1/6 hence the mean is 100
you need to figure out what 20 is in terms of the st. deviation
the variance is npq
But the "standard deviation" is defined as the square root of the variance. Since the variance is "npq", the standard deviation is $\displaystyle \sqrt{npq}= \sqrt{100(1/6)(5/6)}= \frac{10}{6}\sqrt{5}= \frac{4\sqrt{5}}{3}$
20 is $\displaystyle \frac{20}{\frac{4\sqrt{5}}{3}}= \frac{15}{\sqrt{5}}= 3\sqrt{5}$ standard deviations.
we answered that
20 is how many st deviations...
HENCE, divide 20 by $\displaystyle \sqrt{npq}$ and plug that into chebyshev's
AND lower bound on what?
we assume you mean probabilities here.
solve for k and plug into the inequality on either of those links.