Please answer ALL parts legibly or type. And upload images one by one. Thank you

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Please answer ALL parts legibly or type. And upload images one by one. Thank you

PROBLEM C (3 pt) Three members of a private country club were nominated for the office of president The probability that Mr.Adams will be elected is 0.3, the probability that Mr.Brown will be elected is 0.5 and the probability that Mr.Cooper will be elected is 0.2. Should Mr. Adams be elected, the probability for an increase in membership fees is 0.8 Should Mr.Brown or Mr.Cooper be elected, the corresponding probabilities for an increase in fees are 0.1 and 0.4. Find the probability that the membership fees will increase after the election of the new president Suggestion. For convenience you may denote by A, B, C the events that, respec- tively, Mr.Adams, Mr.Brown, Mr.Cooper wl be elected PROBLEM D (6 pt) Refer to Problem C. Unfortunately, we do not know which of the three candidates was elected. What we do know is that after the election the membership fee has increased. Find the posterior probability that the new president of the club is a) Mir Adams; b) Mr. Brown; c) Mr. Cooper. PROBLEM E 1+2+2-5 pt) There are three coins: one is fair (has a head on one side and tails on the other side), the second has heads on both sides, and the third has tails on both sides. A blindfolded man chooses a coin at random and flipsit 1) What is the prior probability P(second) that the second coin will be chosen? 2) What is the probability P(Head) that a head will appear on the upper side of the flipped coin after the experiment is performed? 3) Assume a head has appeared on the top side of the coin after the experiment was performed. What is the posterior probability P(scondHead) that on the bottom side is also a head, i.e. that the second coin was really chosen by the blindfolded man Answer to question 3):

Explanation / Answer

Problem C :

Defined the events

A : Mr . Adams will be elected

B : Mr. Brown will be elected.

C: Mr. cooper will be elected.

P(A) = 0.3, P(B) =0.5 and P(C) = 0.2

I.e Events A, B and C are exhaustive events and they form a partition of sample space

Define the new event

D : membership will increased

P(D/A) = 0.8 ,P(D/B) = 0.1 and P(D/C) = 0.4

By Baye's theorem

P(D) = [ P(A) * P(D/A) ] +  [ P(B) * P(D/B) ] + [ P(C) * P(D/C) ]

= 0.3 *0.8 + 0.5 *0.1 + 0.2 *0.4

= 0.24 + 0.05 + 0.08

=0.37

P( memebership fee will increase) =0.37

Problem D :

P( Mr. Adam will be elceted / membership fee has increased) =P(A/D)

by Baye's Theorem

P(A/D) =  [ P(A) * P(D/A) ] / {  [ P(A) * P(D/A) ] +  [ P(B) * P(D/B) ] + [ P(C) * P(D/C) ]}

P(A/D) = 0.24 /0.37

P(A/D) = 0.6486

P( Mr. Brown will be elceted / membership fee has increased) =P(B/D)

by Baye's Theorem

P(B/D) =  [ P(B) * P(D/B) ] / {  [ P(A) * P(D/A) ] +  [ P(B) * P(D/B) ] + [ P(C) * P(D/C) ]}

= 0.05/0.37

=0.1351

P(B/D) = 0.1351

P( Mr. Cooper will be elceted / membership fee has increased) =P(B/D)

by Baye's Theorem

P(C/D) =  [ P(C) * P(D/C ] / {  [ P(A) * P(D/A) ] +  [ P(B) * P(D/B) ] + [ P(C) * P(D/C) ]}

= 0.08/0.37

=0.2162

P(C/D) = 0.2162

Problem E:

Define the event

A : first coin will be selected.

B : Second coin will be selected.

C : Third coin will be selected.

P(A) =P(B) =P(C) = 1/3

Let H : the head will be appear on the upper side of the filpped coin.

P(H/A) =1/2 , P(H/B) =1 and P(H/C) =0

1) P(Second coin will be choosen) =P(B) = 1/3

2) P( head will be appear on the upper side of the filpped coin.) =P(H)

P( H) = [ P(A) * P(H/A) ] + [ P(B) * P(H/B) ]+[ P(C) * P(H/C) ]

= 1/3 * 1/2 + 1/3 *1 + 1/3 *0

= 3/6 =1/2

P(H) =1/2

3) P( second Coin will be select / The coin shows head) = P(B/H)

By Baye's theorem

P(B/H) = P(B) *P(H/B) / { [ P(A) * P(H/A) ] + [ P(B) * P(H/B) ]+[ P(C) * P(H/C) ]}

P(B/H) = (1/3 *1) / ( 1/2)

=2/3

P( second Coin will be select / The coin shows head) = P(B/H) = 2/3

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