Tom K. answered • 07/09/20
Knowledgeable and Friendly Math and Statistics Tutor
This is easily shown.
P(AB) = 1 - P(~A U ~B) >= 1 - min(1,P(~A2) + P(~B)) = 1 - min(1,1 - P(A) + 1 - P(B)) = max(0, a + b - 1)
Then, P(AB - C) = P(AB - ABC - ~(AB)C) = P(AB) - P(ABC) >= P(AB) - P(C) >= max(0, a + b - 1) - c >= a + b - 1 - c = a+ b - c - 1
P(AB - C) >= a + b - c - 1
Tom K.
minimum and maximum07/09/20
Tom K.
since the total probability has to be <= 1, the min is needed. the negative of a min creates a max. - min(a, b) = max(-a, -b); as P(~A U ~B) <= min(1, P(~A) + P(~B)), - P(~A U ~B) >= - min(1, P(~A) + P(~B))07/09/20
Tom K.
You actually could have solved this without using the min. You could have written P(AB) = 1 - P(~A U ~B) >= 1 - (P(~A) + P(~B)) = 1 - (1-a + 1 - b)) = a + b - 1, or P(AB) >= a + b - 1, so P(AB - C) = P(AB - ABC - ~(AB)C) >= P(AB) - P(C) >= a + b - 1 - c = a + b - c - 1, or P(AB - C) >= a + b - c - 1 It is just nice to keep this in mind07/09/20
Ysabelle C.
By which theorems or properties do the minimum and maximum represent, Sir? I only know the basic axioms and theorems like De Morgan's Law, Finite Additivity, etc. My professor tasked me to state justifications for each step of the proof however, I don't know what theorem/justification I should provide for the min and max.07/09/20
Tom K.
The properties you are relying on are that P(A U B) <= P(A) + P(B) and 0 <= P(A) <= 1, basic properties of probability. min and max come from the fact that if x <= a and x <= b then x <= min(a,b). This is a basic property of number. Similarly, when you subtract a min , we are then looking at max (think of how dividing by a negative number reverses signs). You can actually do the proof that I sent second, I believe, as it doesn't look like you had to use the fact that P(A U B) <= 1. I wrote the P(AB - C) as I did because P(AB - C) = P(AB) - P(ABC), and as P(ABC) <= C, P(AB) - P(ABC) >= P(AB) - P(C). (Note the switch in the inequality symbol)07/09/20
Ysabelle C.
Got it, Sir. Thank you, Sir Tom, for taking the time to answer my questions! I don't have anything to offer yet you answer my queries for free. Thank you Sir for helping students like me. My prayers are with you. All the best, Sir!07/10/20
Ysabelle C.
Thank you Sir! What do min and max represent?07/09/20