AMS 5 Lecture 14 582017 (8:009:05) Law of Averages Law of large numbers o If we toss a (fair) coin n times o we will probably see about n2 heads The number of heads in n tosses = n2 + chance error () percentage of heads in n tosses = 50 + chance error () Law of averages does not say o We will see exactly n2 heads o We will definitely see between n2 + C.E. () heads o We will see between 50 + C.E heads. o anything about what will happen on the next toss. [0] [1] box o Box of tickets: [0]s and [1]s proportion of [1]s = p of [1]s = p*100 o Law of averages for [0] [1] box: If n tickets are drawn at random with replacement, then we will probably see about p*n [1]s. The of [1]s in n draws is probably in the range p*n + CE() Observation: When drawing from a [0] [1] box, then of [1]s we observe = sum of the draws. o Sum of draws, in general Box of numbered tickets Tickets are drawn at random with replacement o Question: If n tickets are drawn, what is the expected sum of the draws? Individual ticket drawn ~ avg(box) (best guess) Sosum of draws (n) ~ n *avg(box) o Definition: Expected value for the sum of n drawsE U (sum) = n*avg(box)

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