A and B each have two unbiased four-faced dice, the four faces being numbered 1, 2, 3 and 4. Without looking, B tries to guess the sum x of the numbers on the bottom faces of A’s two dice after they have been thrown onto a table. If the guess is correct B receives x^2 euros, but if not he loses x

Question

A and B each have two unbiased four-faced dice, the four faces being numbered 1, 2, 3 and 4. Without looking, B tries to guess the sum x of the numbers on the bottom faces of A’s two dice after they have been thrown onto a table. If the guess is correct B receives ${x}^{2}$ euros, but if not he loses x euros.
Determine B’s expected gain per throw of A’s dice when he adopts each of the following strategies:
(a) he selects x at random in the range 2 ≤ x ≤ 8;
(b) he throws his own two dice and guesses x to be whatever they indicate;
(c) he takes your advice and always chooses the same value for x. Which number would you advise?

Accepted Answer

We first calculate the probabilities p(x) and the corresponding gains [latex]g(x) = p(x){x}^{2} − [ 1 − p(x) ]x[/latex] for each value of the total x. Expressing both in units of 1/16, they are as follows:

Table (1)

(a) If B’s guess is random in the range 2 ≤ x ≤ 8 then his expected return is

[latex]\frac { 1 }{ 16 } \frac { 1 }{ 7 } (−26 − 24 − 4 + 40 + 30 + 0 − 56) = −\frac { 40 }{ 112 }= −0.36 euros[/latex].

(b) If he picks by throwing his own dice then his distribution of guesses is the same as that of p(x) and his expected return is

[latex]\frac { 1 }{ 16 } \frac { 1 }{16 }[1(−26) + 2(−24) + 3(−4) + 4(40) + 3(30) + 2(0) + 1(−56)][/latex]

[latex]=\frac { 108 }{ 256 }= 0.42 euros[/latex].

(c) As is clear from the tabulation, the best return of 40/16 = 2.5 euros is expected if B always chooses ‘5’ as his guess. Of course, you should not advise him but offer to take his place!

Question : A and B each have two unbiased four-faced dice, the four fac...

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