Q: In the Gambler’s Ruin recurrence, why do we assume Pi = r^i?
A: The recurrence contains shifted indices i +1, i, and i-1, so we look for a sequence whose form is preserved when the index is shifted.
Math:
Q: What happens when we shift the index by 1?
A: A geometric sequence remains a geometric sequence after shifting, differing only by a constant factor.
Math:
Math:
Q: Is P_i=r^i derived from a known probability value?
A: No. It is a trial solution (ansatz) used to discover which values of r satisfy the recurrence relation.
Q: If I know a probability is 0.4, should I write r^i=0.4 and solve for r?
A: No. During the characteristic equation step, neither P_i nor r is known. The recurrence determines the possible values of r.
Q: Why does the assumption P_i=r^i make the algebra easier?
A: Because every term in the recurrence contains a common factor r^i, which can be factored out and cancelled.
Math:
Math:
Q: What does r represent intuitively?
A: r is the ratio between consecutive terms of the sequence.
Math:
If r>1, the sequence grows; if 0<r<1, it decays; if r=1, it remains constant.
Q: What is the key intuition behind the characteristic equation method?
A: We choose a form that is preserved by the operation appearing in the equation. Since recurrence relations involve index shifts, geometric sequences r^i are the discrete analogue of exponentials in differential equations.
Math:
This is why the presence of i+1, i, and i-1 naturally suggests trying a geometric sequence.