# Bernoulli distribution vs the probability mass function

What is the difference between the two? They seem to mean the same thing to me. The probability mass function can be used to find the probability of getting a tail from a coin flip: X{1 tail}=P(X)=P(H),P(T)=1/2

The Bernoulli distribution equation goes like: (1/2)^1(1-.5)^(1-1)=1/2 because p(success) is 1/2, and p(failure) is 1-(1/2).

Are they the same thing? If not what types of problems would one shine over the other? I know that we can use the Bernoulli distribution as the basis of the Binomial distribution to find the number of tails in n trials but is that it? Is the difference that the bernouli distribution equation can be used to find n tails until a head comes up? Are the bernoulli and the probability mass function just used for the probability of a fixed random variable X, which is a subset of the bionomial distribution?

A probability mass function is a function for a discrete random variable which returns probabilities. We denote this by $Pr(X=x)$ which means the probability of the random variable $X$ being $x$. Some examples:
\begin{align} P(X=x)&=\frac{e^{-\lambda}\lambda^x}{x!}, \quad x=1, 2, \dots \tag{Poisson}\\ P(X=x)&={n \choose x}\theta^x(1-\theta)^{n-x}, \quad x=0, 1, \dots, n \tag{Binomial}\\ P(X=x)&=\frac{1}{10}, \quad x=1, 2, \dots, 10 \tag{Uniform}\\ P(X=x)&=\theta^x(1-\theta)^{1-x}, \quad x=0, 1 \tag{Bernoulli} \end{align}
As you point out, if you let $n=1$ in the binomial then you have a Bernoulli distribution. Furthermore, with $\theta=1/2$ you get your example of coin flips. But $\theta$ can be anything between 0 and 1.