Suppose that you are flipping 5 biased coins independently from one another. The

Question

Suppose that you are flipping 5 biased coins independently from one another. The probability of a coin landing HEAD is 0.6, and the probability of a coin landing TAIL is 0.4.

(a) What is the probability that your ve tosses result in HTHTH, in that specific order?
(b) How many ways are there to land 3 H's and 2 T's? For example, one way is to have HTHTH, i.e. first coin lands H, second coin lands T, etc. Please list all possibilities.
(c) What is the probability that your five tosses result in 3 H's and 2 T's?
(d) Suppose that you are flipping n biased coins independently from one another. The probability of a coin landing HEAD is p, and the probability of a coin landing TAIL is 1 - p. What is the probability that you land k H's and (n - k) T's?
(e) Let the random variable X denote the number of H's you will get in n coin flips, where the probability of landing H is p. X is known as a binomial random variable,and is often denoted as X ~ Bin(n, p). Find E[X] and Var[X].

Explanation / Answer

(a)

probability = 0.6 * 0.4 * 0.6 * 0.4 *0.6 = 0.03456

(b)

The total possibilities are

(1) HHHTT
(2) HHTHT
(3) HHTTH
(4)HTHHT
(5)HTHTH
(6) HTTHH
(7) THHHT
(8) THHTH
(9) THTHH
(10) TTHHH

(C)

Probability = no.of possibilities * 0.6^3 * 0.4^ 2 = 10 * 0.03456 = 0.3456

(d)

Probaility = nCk * p^k * (1 -p)^(n-k)

(e)

Mean = (0 * (1-p)^n + 1*p*(1-p)^(n-1) + 2 * p^2 * (1-p)^(n-2) +........+ n *p^n)/(n+1)

E(x) = [n0 i * p^i * (1-p)^(n-i)]- (n+1)* mean

V(x) = n0 (i * p^i * (1-p)^(n-i)]- mean)^2

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