Question

STATS... Based on conditional probibility, if P(B) > 0, show P(A|B) = 1 - P(A'|B) ... need sum help if sum1 has taken STAT101

Answer 1

The event A will either happen, or it won't happen. One or the other must happen (but not both at the same time)

So saying P(A or A') means you say

P(A) + P(A')

and this is equal to 1 (which represents 100%) since one or the other (but not both) is guaranteed to happen, so

P(A) + P(A') = 1

solve for P(A) to get

P(A) = 1 - P(A')

Answer 2

Similarly

If some other event happens, call this event B, and it affects event A

then event A will either happen or it won't happen (sure it may change the probability of each, but one or the other must be true)

So this means

P(A|B) + P(A'|B) = 1

and solving for P(A|B) gives us

P(A|B) = 1 - P(A'|B)