. Toss n coins. We saw this before, but assumed that the coins are fair. Now we donot. The sample space isΩ = {0, 1}n = {ω = (ω1, . . . , ωn) : ωi = 0 or 1 for each i ≤ n}.Further we assign pω = α(1)ω1. . . α(n)ωn. Here α(j)0and α(j)1are supposed to indicate the probabilitiesthat the jth coin falls tails up or heads up, respectively. Why did we take the product of α(j)· s andnot some other combination? This is a non-mathematical question about what model is suitedfor the given real-life example. For now, the only justification is that empirically the above modelseems to capture the real life situation accurately.In particular, if the n coins are identical, we may write p = α(j)1(for any j) and the elementaryprobabilities become pω = pPi ωiqn−Pi ωi where q = 1 − p.Fix 0 ≤ k ≤ n and let Bk = {ω :Pni=1 ωi = k} be the event that we see exactly k heads outof n tosses. Then P(Bk) =nkpkqn−k. If Ak is the event that there are at least k heads, thenP(Ak) = Pn`=kn`p`qn−`


standard deviation
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