Poisson distribution equation

Question

This is probably a very simple and silly question to ask, but I just don't understand the steps for b). I don't quite understand where the negative (-) sign came from? Could somebody please shed some light on the rationale behind b)? Thank you.

https://i.stack.imgur.com/cZxSd.png

Answer 1

$X$ takes on values in $\{0,1,2,\ldots,\}$. The complement of the event $X \ge 1$ is $X=0$, so $P(X \ge 1) = 1-P(X=0)$.

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Edit for clarity: If $A$ is some event, and $A^c$ is its complement (i.e., "not $A$"), then you should have a result somewhere in your textbook that says $$P(A)+P(A^c)=1.$$ Here, the event that you are concerned with is $X \ge 1$. Its complement would be "not $X \ge 1$," which is $X<1$. However, since $X$ is a Poisson random variable, its value must be a non-negative integer, so the event $X<1$ is equivalent to the event $X=0$. So, $$P(X \ge 1)+P(X=0)=1.$$