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Question

.oooo Sprint LTE 10:36 PM a bb.uccs.edu (c) Sis thousand players each pay $10,000 to cer the world scrics of polcz Ec starts the sournament with $10K in chigs, and the play no limit Hold"emsoowissal chips. The top 600 players receive prize money accuedeg to ktfai they boad. The airline assigns the oeder of boanding acceeding to the time thehe insp US. "Amy gain by the winner must harm the loser Is this statcmen tuc or fabe En V4. Alice, Bob, and Canfwws are bueodarng rous ”they do idetople pm Eachof wins if two heads and one tail lnd and Confacis wis if onc head and twotails The q n and he winnar reeives a net gain of $2 (53-51 entry fee What is the peobability that Alioe will win What is Alice's cpected payoff (c What isthentuhity that Cathis will win? el Is this a nero sum game? explais S.Whn one player surprises anoeher,this indicates that the player d nohve knowledge ef the nules. Give an example than illstrales that this is a te and aceesamgle hows it in fabe

Explanation / Answer

U4

If p be the probability a player wins, then the probability the player loses is (1 - p) and hence the player’s expected pay-off = E(P)

= (2 x p) + {(- 1 x (1 - p)} = 2p – 1 + p = 3p – 1 ……………………………………… (1)

Back-up Theory

When 3 players flip an unbiased coin each, there are 8 possible outcomes, namely

(HHH), (HHT), (HTH), (HTT), (THH), (THT), (TTH), (TTT). Thus,

There are

2 possibilities where all 3 are H or T;

3 possibilities where 2 are H and 1 is T; and

3 possibilities where 2 are H and 1 is T.

Hence,

P(3H or 3T) = 2/8 ……………………………………………………………(2)

P(2H and 1T) = 3/8 ……………………………………………………………(3)

P(1H and 2T) = 3/8 ……………………………………………………………(4)

Part (a)

Probability Alice would win = P(3H or 3T) = 2/8 = ¼ ANSWER [by (2) above.]

Part (b)

Expected pay-off of Alice = 3(1/4) – 1 = - ¼ or a loss of 0.25 ANSWER [vide (1) and by Part (a), p = 1/4]

Part (c)

Probability Confucius would win = P(1H and 2T) = 3/8 ANSWER [by (3) above.]

Part (d)

Expected pay-off of Confucius = 3(3/8) – 1 = 1/8 or 0.125 ANSWER [vide (1) and by Part (c), p = 3/8]

Part (e)

Probability Bob would win = P(2H and 1T) = 3/8 and hence expected pay-off of Bob

= 3(3/8) – 1 = 1/8 or 0.125

Thus, total expected pay-off of the three players = - ¼ + 1/8 + 1/8 = 0.

Hence, this is a zero-sum game. ANSWER

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