Consider a coin with P(Heads) = 2/ 3 . We toss this coin 100 times (assume that the tosses are independent). Determine the probability that we get exactly 45 tails out of the 100 tosses. First, compute an exact expression (this might require matlab or some similar tool). Next, determine an approximate numerical value of this probability by using the central limit theorem, and the standard Gaussian table.
So I've got my $P[x]=\sum$ $_nC_x (1/3)^x(2/3)^{n-x}$
i.e $B(100,1/3)=N(100/3,200/9)$
Now what next?
We want the probability of $45$ tails. This is given exactly by $\binom{100}{45}\left(\frac{1}{3}\right)^{45}\left(\frac{2}{3}\right)^{55}$. Compute, using appropriate software.
For the normal approximation, note that the number $X$ of tails has not too far from normal distribution, mean $\frac{100}{3}$, variance $\sigma^2=\frac{200}{9}$.
We should get a good approximation to the probability that $X=45$ by finding the probability that a suitable normal $Y$ lies between $44.5$ and $45.5$. The suitable normal $Y$ has mean $\frac{100}{3}$ and standard deviation $\sqrt{\frac{200}{9}}$. Compute in the usual way, using a table of the standard normal.