Is this conditional probability always equal to 1?

Question

Consider $X = \text{number of defective items in bought items}$.

Is the probability that $X \geq a$ given $X = a$, always 1: $P(X \geq a| X = a)=1$.

I was wondering if the above holds because $A$ (event that $X \geq a$) will always occur if $B$ (event that $X = a$) occurs, therefore, making the probability 1.

If not why not?

Any thoughts? Thanks in advance.

Answer 1

The event $X \ge a$ is a superset of $X = a$; i.e., $$(X = a) \subseteq (X \ge a).$$ Therefore, $$\Pr[X \ge a \mid X = a] = \frac{\Pr[(X \ge a) \cap (X = a)]}{\Pr[X = a]} = \frac{\Pr[X = a]}{\Pr[X = a]} = 1$$ whenever $\Pr[X = a] > 0$.