We know how to calculate the probability 
.
Now if we toss a coin m times how many tails should we expect?
One way to estimate the expected number of tails consists in adding the
number of tails multiplied by their probability. In this sum, a number of
tails coming with a large probability will contribute more than a number
of tails coming with a small probability. This sum is called the average
number of tails and can be calculated explicitly.
We just quote the result (without proof):
This result can be rephrased as follows: if you toss a coin m times, you expect in average to get a fraction P(T) of the m tosses to give a tail. You can also calculate the size of typical deviations by taking the square root of the average of the square of the number of tails minus its average. We need to take the square of the differences because if we were averaging over the differences, we would simply get zero. One can show that
After taking the square root, we see that typical fluctuations grow only like the square root of the number of tosses.
Note that these results explain the rule of the thumb used to
estimate the size of the fluctuations in the histograms displayed
for the random number generators. If we have random numbers uniformly
distributed between 0 and 1 and if we divide this interval in
100 bins, the probability for a number to be in a particular bin
is 
while the probability to be in any other bin
is 
. This is like tossing a coin which is very
asymmetric and for which 
. If we generate
m=1000 numbers we expect to have mP(T)=10 numbers in each
bin with fluctuations of order 
.