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For any events $A$ and $B$ with $P(B)>0,$ show that $P(A | B)+P\left(A^{\prime} | B\right)=1$

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Probability Topics

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all right, We're given to events A and B, and we're also given that the probability of these greater than zero And what we want to prove is that the probability of a given B plus the probability of a not given B equals one. And for that, we're just gonna use based there, in which are mathematical representation. Probability of egg oven be which looks like this. All right. So far, so good, we're gonna actually represent ah, the probability of a not given be using based armas. Well, so we'll just sub in some may not send for the A's there. Do the same on the denominator. And keep in mind that the compliment of a knot is just a so this just becomes a probability of a and probability of be given a All right now since these two have the same denominator, adding them is rather simple will end up with this. We'll just add the numerator tze and throw the common denominator on the bottom. So, Streit this Outworld quick all over. All right. And if we step back and look this fraction, we realize, Hey, the numerator and denominator, they're the same. So this equals one. Thus be proven that since the fractions add up to one their probability, the vein given B plus the probability of a not given B equals one mission makes sense. Even given a condition, something is either true or not true. So there you go.

University of California - Los Angeles

Probability Topics