Suppose that two weighted (unfair) dice sum to each possible sum-value with an equal probabil- ity....

 

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Suppose that two weighted (unfair) dice sum to each possible sum-value with an equal probabil- ity. Let us denote the probabilities for the vari- ous sides of the first die by p1, P2, .... P6 and the corresponding probabilites for the second die by q1, 92,... 96 - (a) What is the sample space for the sum of the two weighted dice? (b) Explain why P141 = P66 = and p196 + (c) Now apply the inequality (r - y)² = - 2ry + y? 2 0 for suitable choices of a and y to deduce that p196 + Pe91 2 i: (d) Can two weighted dice sum to each possible sum-value with an equal probability? Suppose that two weighted (unfair) dice sum to each possible sum-value with an equal probabil- ity. Let us denote the probabilities for the vari- ous sides of the first die by p1, P2, .... P6 and the corresponding probabilites for the second die by q1, 92,... 96 - (a) What is the sample space for the sum of the two weighted dice? (b) Explain why P141 = P66 = and p196 + (c) Now apply the inequality (r - y)² = - 2ry + y? 2 0 for suitable choices of a and y to deduce that p196 + Pe91 2 i: (d) Can two weighted dice sum to each possible sum-value with an equal probability?