Binomial distribution p=1

Solution 1:

There's 2 cases to look at here.

If $k < n$, we have that the probability of $k$ successes in $n$ trials will be $0$. This is because the binomial distribution parameter, $p$ refers to the probability of success for each individual trial. In the case that $p = 1$, we have that every single trial will be a success. This means that in $n$ trials we will have $n$ successes and so $k$ being less than $n$ guarantees that the probability of having $k$ successes in $n$ trials is $0$.

On the other hand, if $k = n$, using the logic from above, we can find that the probability of $k$ successes in $n$ trials is 1.

Therefore, the PMF is not $0$, as the entire mass of the function is contained at $k = n$.

However, as stated in Rohan's answer, having $p = 1$ doesn't really agree with the idea of a binomial distribution.