# Thread: Definition of Complement

1. ## Definition of Complement

I'm having trouble understanding the following:

Does Ec mean that all events in the sample space (besides E) necessarily happen, or simply that E does not happen? In other words, if the sample space is S={A,B,C,D,E} does Ec imply A,B,C, and D have occurred, or just that E hasn't occurred?

2. ## Re: Definition of Complement

What your image is implying, is that if you have an event E, visa-vi a probability.

Then if $\displaystyle E$ doesn't happen, you have $\displaystyle E^{c}$.

If you have a set of events $\displaystyle A$, $\displaystyle B$, $\displaystyle C$, $\displaystyle D$ and $\displaystyle E$.

Then $\displaystyle E^{c}$ is simply the case where $\displaystyle E$ doesn't happen.

It doesn't mean that the others necessarily happen, nor does it tells us the individual probability of each.

If $\displaystyle P(E) = 0.2$, then at most, we can say that $\displaystyle P(E^{c}) = 0.8$, or $\displaystyle P(E^{c}) = P(A) + P(B) + P(C) + P(D) = 0.8$.

3. ## Re: Definition of Complement

Perfect. Exactly the clarification I needed.