An Extremely Biased Coin - I

Let $X$ be the number of heads obtained after tossing the coin 100 times. Then $X$ has a binomial distribution. That is, if there were $n$ Bernoulli trials, with the probability of success being $p$ , then the probability of $r$ successes is given by:

$P( X = r ) = \binom { n } { r } \ p ^{ r } ( 1 - p)^{ n - r }$

In the problem, since there are $100$ trials: $n = 100$ ; the probability of success is $p = \frac{ 99 } { 100 }$ ; and we want to find the probability of getting $r = 99$ successes.

$P( X = 99 ) = \binom { 100 } { 99 } \left( \frac{ 99 } { 100 } \right)^{99} \left( \frac{ 1 } { 100 } \right)^{ 1 } \approx 0.3697 \ldots$

Here we will use the useful result Poisson approximation of Binomial distributions , which will simplify the computations enormously. Clearly, number of tails for $n$ tosses follow a Binomial distribution with probability of success $p=0.01$ . Here $n=100, p=0.01, np=1$ . Hence the premises of the above approximation holds and we have :
$\text{Binom}(n,p) \approx \text{Poisson}(np).$ Thus probability of a single tail out of $100$ tosses is approximately $e^{-1} \frac{1^1}{1!} = \frac{1}{e} \approx 0.37$ .