SOLUTION: Show that P[A+B+C] = P[A] +P[B] +P[C] �P[AB]-P[AC]-P[BC] +P[ABC]

Question 363455: Show that
P[A+B+C] = P[A] +P[B] +P[C]
�P[AB]-P[AC]-P[BC] +P[ABC]

Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
We can prove this using the more fundamental law: P[A+B] = P[A] + P[B] - P[AB].
P[A+B+C] = P[A+B] + P[C] - P[(A+B)C] = P[A] + P[B] - P[AB] + P[C] - P[AC + BC] =
P[A] + P[B] - P[AB] + P[C] - (P[AC] + P[BC]- P[[AC][BC]]).
Then P[A+B+C] = P[A] + P[B] - P[AB] + P[C] - (P[AC] + P[BC]- P[[AC][BC]])
=P[A] + P[B] - P[AB] + P[C] - P[AC] - P[BC] + P[[AC][BC]]. But P[[AC][BC]] = P[ABC], hence the result.
Question The chip lasts more than 1000 hrs but fewer than 1500 hrs i.e.
E =(1000, 1500). Let the events A,B,C be defined by A=(5,∞), B=(7, ∞) and C = (0,3) Describe these events in words. Find the events AB, AC and A+B and describe these in words
A = (5, ∞ ) lifetime is greater than 5. B = ( 7 , ∞ ) Lifetime is greater then 7. C = ( 0 , 3 ) Lifetime is not greater than 3. AB = ( 7 , ∞ ) Lifetime is greater than 7. AC is the empty set: The two intervals won't even intersect. (Try graphing each interval on the real number line, for example.) A+B = ( 5 , ∞ ) Lifetime is greater than 5.
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