Difference b/w $p$ and $P(X)$ i.e. output of Binomial Distribution ($BD$)? Is it possible to have $p=1$ but $BD=0$?

Question

I have a confusion with Binomial Distribution. For Binomial Distribution we use the formula:

$P(x) = {n \choose k} \cdot p^x \cdot (1-p)^{n-x}$

Now let's suppose $p=1$ but if we put this $1$ in $P(x)$, we will get $P(x)=0$.

How can we define this situation?

Zulfi.

Answer 1

Um, funny. If $p=1$, then when you repeat the experiment $n$ times, $P(X=n)=1$ and $P(X=k)=0$ for all other $k$. You can't use the formula to calculate $P(X=n)$, but for all other $P(X=k)$ the formula still works since $0^{n-k}=0$ if $k\ne n$.

Edit: If you read Hogg and Craig's "Intro to Mathematical Statistics", you will find that they define the binomial distribution with a restriction of $0<p<1$.