Question

(Expected values) Suppose you roll a fair die and your opponent flips a fair coin simultaneously. You win $4 whenever a Head appears and the number of dots on the top face of the dice is either one or six. For all other outcomes, you lose $1.

(a) (2pts) How many possible outcomes are there? An outcome is the combined result of both coin-flipping and dice-rolling.

(b) (3pts) What is your expected payoff?

(c) (3pts) What is your opponent’s expected payoff? Is the game in your favor?

Answer 1

(a)

When you roll a die any natural number from 1 to 6 can appear and when you flip a coin either Head or Tail will appear.

Hence, Number of possible outcomes = 6*2 = 12 namely (1,H), (2,H) , (3,H) , (4,H) , (5,H) ,(6,H), (1,T), (2,T) , (3,T) , (4,T) , (5,T) ,(6,T).

(b)

I will win $4 if we have the following result after rolling a die and coin toss : (1,H) , (6,H). Hence, Probability of winning = P(W) = 2/12 = 1/6

I will loose $1 if i didnt win => Probability of Loosing = P(L) = 1 - 1/6 = 5/6

Hence, Expected payoff = 4*P(W) - 1*P(L) , Here -1 is because he will loose -1.

Hence, Expected payoff = 4*(1/6) - 1*(5/6) = -0.17

(c)

He will loose $4 if I win and I will win if I have the following result after rolling a die and coin toss : (1,H) , (6,H). Hence, Probability of Loosing = P(L') = 2/12 = 1/6

He will win $1 if i will loose => Probability of his winning = P(W') = 1 - 1/6 = 5/6

Hence, His Expected payoff = 1*P(W') - 4*P(W') , Here -4 is because he will loose -4 if he lost.

Hence, Expected payoff = 1*(5/6) - 4*(1/6) = 0.17

Hence, Expected Payoff = 0.17

As my opponents expected payoff is greater than my expected payoff, hence The Game IS NOT in my favor.