Let = {H,T)10, put p = 1/4 and q = 3/4, and define a probability measure P on th

Question

Let = {H,T)10, put p = 1/4 and q = 3/4, and define a probability measure P on the power set of by P(w) p#H(w)q#1(w) for each outcome w E . Think of as a space of 10 biased but independent coin fiips. Calculate the probabilities of each of the following events. You should use conditional probabilities and the fact that you can look at the probability space as the product of 10 iid Bernoulli(1/4) distributions to make your computations simpler. 3.2. (a) The third flip is the first H. (b) The sixth flip is the second H (c) There are more H occurring in the first 4 flips than in the last 4 flips (d) The number of H flips is 3 or fewer.

Explanation / Answer

1)
third flip is the first H
= (3/4)^2 * (1/4)
= 9/64
b)
sixth flip is the second H
= 5C1 * (3/4)^4 * (1/4) * (1/4)
= 5 * (3/4)^4 * (1/4)^2

d)

P(X = k) = nCk * p^k *(1-p)^(n-k)

= 10Ck * (1/4)^k * (3/4)^(10-k)
P(X < = 3) =

P(X < = 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.775875

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