I was doing an exam review and I have absolutely no idea how to do this question. There is no solution to this so I'm hoping someone here knows how to do this.
The fibonacci numbers are defined as follows: f0=0, f1=1, and fn=fn-1 + fn-2 for n>=2. Let n be a large integer. A fibonacci die is a die that has fn faces. Such a die is fair: If we roll it, each face is on top with the same probability 1/fn. There are three different types of fibonacci dice.
D1: fn-2 of its faces show the number 1 and the other fn-1 faces show the number 4.
D2: Each face shows the number 3
D3: fn-2 of its faces show the number 5 and the other fn-1 faces show the number 2.
Assume we roll each of D1, D2, and D3 once, independently of each other. Let R1, R2, and R3 be the numbers on the top face of D1, D2, and D3, respectively. Determine Pr(R1 > R2) and Pr(R2 > R3) and show that Pr(R3 > R1) = ((fn-1)(fn+1)) / (fn)2