WEBVTT
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all right, we're given this identity to prove the
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probability of a union Be given, See is he
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goes to the probability of a given C plus a
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probability of be given C minus the probability of a
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intersects beep Given c for this, we're gonna use
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to hard probability identities, one of which is the
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multiplication I didn t which is as follows the probability
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of a intersect be It was the probability of a
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given b times the probability of b what we're going
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to do. So we're going to rewrite this probability
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a given B Because the probability a intersect to be
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over probability of B the second rule will need is
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our additional probability of a union be It was the
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probability of a it's probability of B minus the probability
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of a intersects B All right, so let's use
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his modified multiplication rule on the left hand side with
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this identity probability of a union be given. See
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those probability of a union, be intersex, see
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all over the probability of C intersection quote unquote distributes
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. So this because probability of a intersex see union
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be in our sexy over probability. See, now
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we're going to use our addition rule to expand this
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out of little bits of probability of And her sexy
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lost the probability of be her sexy minus the probability
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of, well, normally it be a and her
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sexy intersect be in her sexy. But that just
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is a intersects b intersect. See, we don't
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need to be redundant to repeat ourselves. We're gonna
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separate this into three separate fractions using the same denominator
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. All right as follows. Sorry. That should
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be a minus. Now, we're going to use
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our modified multiplication rule once again. So this becomes
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probability of a given c. This becomes probability of
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being her sexy. And because intersection is associative,
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we can put parentheses wherever he wants. Are gonna
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put prince sees around, are a intersect, Be
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to make this intersect, to be given C,
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and thus we've gotten to the right side of our
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identity. Q e d