WEBVTT
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In a sample space π, the probabilities are shown for the combinations of events π΄ and π΅ occurring.
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Are π΄ and π΅ independent events?
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What can we say about the probability of independent events?
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The probability of π΄ and π΅ occurring equals the probability of π΄ times the probability of π΅.
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We need to check if the probability of π΄ and π΅, the probability in the middle, is equal to the probability of π΄ times the probability of π΅.
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But before we do that, notice how each of these probabilities have a different denominator.
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In order to work with these and compare them accurately, we need to have a common denominator.
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The least common multiple from five, 10, and 15 will be 30.
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To go from 15 to 30, weβll multiply by two.
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If we multiply by two in our denominator, we need to multiply by two in our numerator.
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Seven times two equals 14.
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Five times six equals 30.
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If we multiply by six in the denominator, we need to multiply by six in the numerator.
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One times six equals six.
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10 times three equals 30.
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And one times three equals three.
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~~Seventeen~~ [seven] fifteenths equals fourteen thirtieths.
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One-fifth equals six thirtieths.
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And one-tenth equals three thirtieths.
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The probability of π΄ and π΅ will be the intersection of the probabilities π΄ and π΅.
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Thatβs six thirtieths.
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The probability of π΄ is a little bit trickier.
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The probability of π΄ is equal to fourteen thirtieths plus six thirtieths.
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Itβs equal to the two probabilities found in the π΄ circle.
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Fourteen thirtieths plus six thirtieths equals twenty thirtieths.
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And we need to multiply that by the probability of event π΅.
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Event π΅ will be six thirtieths plus three thirtieths, both of the probabilities found in the π΅ circle.
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Adding six thirtieths and three thirtieths, we get nine thirtieths.
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To multiply fractions, we multiply their numerators.
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20 times nine equals 180.
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30 times 30 equals 900.
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We can simplify by dropping those zeros.
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And then, we could say that 90 is divisible by 18.
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We can divide the numerator and denominator by 18.
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18 divided by 18 equals one.
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90 divided by 18 equals five.
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We now need to ask the question is six thirtieths equal to one-fifth?
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Remember that we already calculated what one-fifth is.
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We know that one-fifth equals six thirtieths.
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And because the probability of π΄ and π΅ is equal to the probability of π΄ times the probability of π΅, we can say that, yes, these are independent events.