WEBVTT
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An experiment produces the discrete random variable π that has the probability distribution shown.
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If a very high number of trials were carried out, what would be the likely mean of all the outcomes?
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The law of large numbers, also sometimes known as the law of averages, says that the mean of the results from a very high number of trials tends to the expected value.
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Here thatβs πΈ of π₯.
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As the number of trials π tends to β, we say that the mean is equal to πΈ of π₯: the expected value of π₯.
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The formula we need to know for the expected value of π₯ is given by the sum of π₯ multiplied by π of π₯.
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Itβs the sum of each of the possible outcomes multiplied by the probability of this outcome occurring.
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So letβs substitute what we have into this formula.
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π₯ multiplied by π of π₯ for the first column is two multiplied by 0.1.
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For the second column, itβs three multiplied by 0.3.
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For the third column, four multiplied by 0.2.
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And for the fourth and final column, thatβs five multiplied by 0.4.
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Two multiplied by 0.1 is 0.2, add 0.9, add 0.8, and five multiplied by 0.4 is 2.0.
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So we add 2.0 at the end of this line.
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The sum of these values is 3.9.
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So the expected value of π is 3.9.
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And we said that for a very high number of trials, we could call that the mean.
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Now we can look at our table to check whether this answer is likely to be correct.
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Since the possible values for π₯ are two, three, four, and five, and 3.9 is a little over halfway between two and five, 3.9 is likely to be correct for the mean of this probability distribution.