That Bernoulli distribution was derived as follows:

You start at 0 on a number line, and for each step you can go one interval to the left or to the right. To reach any point,

, in

steps, you have to take

steps to the right and

steps to the left. With that said:

Through substitution:

The total number of possible paths by which to arrive at

in

steps is:

If we want to get to any point,

, in

moves, then we know how many moves we must take to the right,

, and how many to the left,

. If

represents successes and

represents failures then:

Which equals:

To reiterate my question, how is it that the distribution above can go from being a Bernoulli distribution to a Gaussian distribution as

approaches infinity?...assuming that the above distribution is in fact a Bernoulli distribution and I'm not completely wrong about everything I just said.