Assume Town A has three parties, Party B, Party C and Party D, with n_1, n_2 and

Question

Assume Town A has three parties, Party B, Party C and Party D, with n_1, n_2 and n_3 citizens in them, respectively. Assume, for people in B, they will always support the motion by the government, while, for people in C, they will always against for the government. However, those in D will support the policy by the government randomly with probability p and will make their decision independently for each policy. If a citizen is randomly chosen who showed the support to the government twice successively, find the probability he/she will support the government again.

Explanation / Answer

Probability of a person being of party B = n1/(n1+n2+n3)

Probability of a person being of party C = n2/(n1+n2+n3)

Probability of a person being of party D = n3/(n1+n2+n3)

Probability of someone supporting govt policy for the third time = (n1xn1xn1 + pxn3 x pxn3 x pxn3)/(n1+n2+n3)3

P(someone supporting govt policy twice)

= (n1xn1 + pxn3 x pxn3)/(n1+n2+n3)2

P(someone supporting govt again if he/she had supported twice successively)

= [(n13+ p3xn33)/(n1+n2+n3)3]/[(n12+p2x n32)/(n1+n2+n3)2]

= (n13 + p3 x n33)/[n1+n2+n3)x(n12+p2xn32)

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