# Definition of Complement

• Jun 8th 2013, 11:09 PM
divinelogos
Definition of Complement
I'm having trouble understanding the following:

Attachment 28554

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?
• Jun 9th 2013, 12:18 AM
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\$.
• Jun 9th 2013, 11:18 AM
divinelogos
Re: Definition of Complement
Perfect. Exactly the clarification I needed.