WEBVTT
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case, So I'm going to pose another sure that
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question. So we want to show that if the
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probability of be given A is bigger than the probability
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be than the probability of being compliment given A it's
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less than the probability of beat, uh, probably
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the compliment of being so to do this and so
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will use the hint that we've been given. So
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what the hint is we want at both sides of
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this question here by the probability of being cobbled,
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eh? So, uh, but size Let me
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call this equation one by probability, off be compliment
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given A and that should give us the probability of
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be given a plus the probability o b compliment given
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a, um, which is bigger than the probability
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o. B plus the pro Billy off. Be
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compliment, given a now from the previous exercise from
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X to size 61. So this is from exercise
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61 off the same chapter in section we have that
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the left hand side equal to one. So what
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happens that years you got one bigot than the probability
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off B, plus the probability off. Be compliment
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, given a and then subtract by sides by the
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probability off B So that gives you one month is
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the probability of B Plus the problem is bigger than
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the probability off. Be compliment given. Hey,
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So what that implies, because the left hand side
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is pretty much just the probability off the compliment of
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B. This implies that the probability off be compliment
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, given a has to be less than because we
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just flipping ever be reading a full brought to left
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will be more than the probability off big compliment.
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So that pretty much concludes the proof. An alternative
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way to get at this a lot quicker is the
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following. So if the pros are beast of probability
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of, um, we're gonna run this probably so
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if the probability oh, be given a he's greater
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than the probability off B then the probability of be
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compliment given a So this is from exercise 60 water
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get We'll just be one minus the probability off be
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given. Hey, now we know that the probability
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of be given a is bigger than the probability be
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so this should imply that one minus the probability off
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be given A has to be less than one,
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minus the probability off, baby Because when you take
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the negative up birth sides, the sign here flips
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and then adding wanted both sides does not change the
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equality. So this will be blue less than one
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minus the probability off B, which is the probability
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off be compliment. So you have that this is
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up less than that. This is the quick way
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of going about things older. Doing it this way
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in going in this way follows the hints that we've
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been given for this.