Sunday, 8 July 2012

Importance of Central Limit Theorum

This question would signify the usefulness of CLT

A coin is tossed 100 times. I would pay you 1000 if Head appears less than 40 times otherwise you pay me 100. Would you be willing to play?

2 comments:

  1. the deviation from the mean is sqrt(n)/2, where n = no. of toss... now if n_h = no. of time head appears, then p(H) = n_h/n +/- sqrt(n)/2n ... so putting p(H) = 0.5, n = 100, we get n_h = 50 +/- 5 = 45 - 65 range.... so it's safe to play.....

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  2. Mean = np = 100*0.5 =50
    Std = sqrt(np(1-p)) = sqrt(100*0.5*0.5) = 5
    Assuming it to be a Normal Dist by CLT
    P(45<X<55) = 0.68
    P(40<X<60) = 0.95
    Thus the probability of heads lesser than 40 would be as low as (1-0.95)/2 = 0.0275
    Expected payoff = 0.0275*1000 - 0.975*100 = -70
    Thus, one should not play

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