SOLUTION: A is an event, and Ac is the complement of A. Which of the following statements are true? (There are more than one true statements) SHOW WORK (a) P(A AND A^c )=0 (b) P(A AND A^

Algebra ->  Probability-and-statistics -> SOLUTION: A is an event, and Ac is the complement of A. Which of the following statements are true? (There are more than one true statements) SHOW WORK (a) P(A AND A^c )=0 (b) P(A AND A^      Log On


   

Question 1094176: A is an event, and Ac is the complement of A. Which of the following statements are true? (There are more than one true statements) SHOW WORK
(a) P(A AND A^c )=0
(b) P(A AND A^c )=1
(c) P(A OR A^c )=0
(d) P(A OR A^c )=1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

A is some event while A^c is the complement of said event.

What this means is that if event A doesn't happen, then surely A^c will happen.
The same can be said in reverse: If A^c doesn't happen then event A will happen.

These two events are completely opposite in nature.
So P(A and A^c) = 0 because it's impossible to have BOTH happen at the same time.
So choice A is true making choice B to be false.

Similarly, P(A or A^c) = 1 because either one or the other must happen, giving us 100% certainty of this compound event.
That makes choice D true and choice C false.