6. (12 points) A box contains three coins. One of the coins is a two-headed coin

Question

6. (12 points) A box contains three coins. One of the coins is a two-headed coin (that is both the sides are head), the second is a fair coin, and the third is a biased coin with P (head) = p. One of the coins is picked at random and flipped. a. Find the probability that the coin shows head. What is its maximum value? b. If the coin shows head, find the probability that it is the fair coin. What are its maximum and minimum values? [Hint: Let A p.l coin shows heads; B1 = coin is two-headed; B2-coin is fair; B3-coin is biased with P(head) =

Explanation / Answer

a) Probability of head =P(A) =P(B1)*P(A|B1)+P(B2)*P(A|B2)+P(B3)*P(A|B3)

=(1/3)*(1)+(1/3)*(1/2)+(1/3)*p =(1/3)*(3/2+p)

maximum value of P(A) will occur when p=1

hence maximum value P(A)max =(!/3)*(3/2+1) =5/6

b) probability of fair coin given head =P(B2|A) =P(B2)*P(A|B2)/P(A) =(1/3)*(1/2)/((1/3)*(3/2+p))

=1/(3+2p)

maximum value when p= 0 ; P(B2|A)max =1/3

minimum value when p= 1 ; P(B2|A)min =1/(3+2*1) =1/5

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