assume that an experiment is conducted and that the outcome is considered o be either success or a failure. Let p denote the probability of success. Define X to be 1 if the experiment is success and 0 if it is a failure. X is said to have a point binomial or a Bernoulli distribution with parameter p.
How do i argue that X is a binomial random variable with n=1 and find density for X ?
If $X\sim \text{Bin}(n,p)$, then $$P(X= k) = \binom{n}{k}p^k (1-p)^{n-k}$$ But the only possible values of $X$ are $0$ or $1$ and $n= 1$, hence $$P(X = k) = \binom{1}{k}p^k(1-p)^{1-k}.$$ Check that this has the same distribution (table) as a Bernoulli trial with probability $p$.