Question

Your instructor randomly chose a coin with probability 0.5 and asks you to decide which coin he chose according to the outcome of 3 tosses: Tossing coin 1 yields a head with a probability P(X1 = H) = .3 (and tail with P(X1 = T) = .7). Tossing coin 2 yields a head with a probability P(X2 = H) = .6 (and tail with P(X2 = T) = .4). You earn $1 if you correctly guessed the coin and $0 otherwise. Design the optimum decision rule and estimate your average earning.

Answer 1

ANSWER::

C1: Coin 1

C2: Coin 2

After tossing a Coin outcome will be head or tail.

Case:1 Heads Comes up, it will came by C1 or C2.

Calculating Their respective probabilities by using Bayes Theorm

P(C1/H) = P(H/C1) P(C1) / { P(H/C1) P(C1) + P(H/C2) P(C2) }

= (.3*.5)/{(.3*.5)+(.6*.5)

= 1/3

Similarly P(C2/H) = 2/3

Ife he gussed that coin is tossed by C1 then Average earning is = $1(1/3)+$0(2/3)

= $1/3

If he gussed that coin is tossed by C2 then average earning is = $1(2/3)+$0(2/3)

= $2/3.

Case 2: If tails Comes up, it will came by C1 or C2.

Calculating their respective probabilities by using Bayes Theorm.

P(C1/T) = P(T/C1) P(C1) / { P(T/C1) P(C1) + P(T/C2) P(C2) }

= (.7* .5) / { .7*.5 + .4* .5}

= 7/11

Similary P( C2/T) = 5/11

If he gussed that coin is tossed by C1 then their average winning is = $1(7/11)+ $0(5/11)

= $7/11.

If he gussed that coin is toseed by C2 then their average winning is = $1(5/11)+ $0(7/11)

= $5/11.

So. Optimal way is if coin shows HEAD then he can choose by Coin C2 and if TAILS shows up then he can choose Coin C1.

Average Earning in a whole is

$1/3 + $2/3 + $7/11 + $5/11 = $2.

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