Distribution of the probability for the number of heads in 100 throws of a coin
For large values of n, the variable x which measures the number of successes
fits the normal curve increasingly well. The larger the value of n the better the
approximation and tossing the coin 100 times qualifies as large. Now let’s say we
want to know the probability of throwing between 40 and 60 heads. The area A
shows the region we’re interested in and gives us the probability of tossing
between 40 and 60 heads which we write as prob(40 (≤x ≤ 60). To find the
actual numerical value we need to use precalculated mathematical tables, and
once this has been done, we find prob(40 ≤ (x ≤ 60) = 0.9545. This shows that
getting between 40 and 60 heads in 100 tosses of a coin is 95.45%, which
means that this is very likely.
The area left over is 1 – 0.9545 which is a mere 0.0455. As the normal curve
is symmetric about its middle, half of this will give the probability of getting
more than 60 heads in a 100 tosses of the coin. This is just 2.275% and
represents a very slim chance indeed. If you visit Las Vegas this would be a bet
to leave well alone.

the condensed idea

The ubiquitous bell-shaped curve