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Instructor: Welcome back everybody.
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Many scientific papers rely on conducting experiments
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or surveys.
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They often provide summarized statistics
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we use to analyze and interpret
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how certain factors affect one another.
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An example would illustrate this better.
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Imagine you conducted a survey
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where 100 men and women of all ages were asked
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if they eat meat.
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The results are summarized in the table on the screen.
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We see 15 of the 47 women that participated are vegetarian
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as are 29 out of the 53 men.
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Now, if A represents being vegetarian
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and B represents being a woman
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then P of A,
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given B
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and P of B,
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given A
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express different events.
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The former equals 15 over 47
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and represents the likelihood of a woman being vegetarian.
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While the latter equals 15 over 44
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and expresses the likelihood of a vegetarian being a woman.
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Since 15 over 44 is greater than 15 over 47,
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it is more likely for a vegetarian to be female
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than for of a woman not to eat meat.
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That's quite interesting
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and shows you that in probability theory
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things are never straightforward.
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Now that you understand the distinctions between
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the different conditional probabilities of two events,
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we can introduce an important concept,
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the law of total probability.
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Okay.
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Imagine A is the union of some finitely many events,
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B1, B2 and so on.
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This law dictates that the probability of A is
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the sum of all the conditional probabilities of A
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given some B multiplied
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by the probability of the associated B.
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To clear any confusion.
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Let us go back to the survey example.
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The probability of being vegetarian
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equals the sum of the probability of being a vegetarian
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given the person is male
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times the probability of being male
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and the probability of being a vegetarian
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given the person is female
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multiplied by the probability of being female.
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If we plug in values from the table,
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we get the probability of being vegetarian
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equals 29 over 53,
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times 53 over 100,
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plus 15 over 47,
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times 47 over 100.
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That's 44 over 100
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or 0.44.
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Therefore, according to the survey
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there is a 44% chance of someone being vegetarian.
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We can refer back to the table and confirm
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the probability of being vegetarian is indeed 0.44.
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So our answer checks out.
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Great.
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In the next video
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we are going to introduce the Additive Law
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and the multiplication rule.
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Thanks for watching.
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