Ysabelle C.

asked • 07/09/20

Prove the statement.

If P(A) = a, P(B) = b, P(C) = c then P(AB-C) > a+b-c-1.

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Tom K. answered • 07/09/20

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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

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Ysabelle C.

Thank you Sir! What do min and max represent?
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07/09/20

Tom K.

tutor
minimum and maximum
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07/09/20

Tom K.

tutor
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))
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07/09/20

Tom K.

tutor
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 mind
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07/09/20

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.
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07/09/20

Tom K.

tutor
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)
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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!
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07/10/20

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