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GN
Numerade Educator

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Problem 62 Medium Difficulty

If $P(B | A)>P(B)$ show that $P\left(B^{\prime} | A\right)<P\left(B^{\prime}\right) .$ [Hint: Add $P\left(B^{\prime} | A\right)$ to both sides of the given inequality and then use the result of the previous exercise.

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Video Transcript

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