The gambler's ruin

Start with i units. Each round win one unit with probability p or lose one with probability 1 − p. Stop when you reach the target N or go broke. How likely are you to reach the target, and how long does it take?

From the slides: Additional material and Probability and what happens when you learn something.

Set up the game

Start with i Target N Chance of winning a round p

Games played

Reached the target

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Average length (rounds)

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Longest game

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If you must gamble, gamble boldly

Ravi has ₹50 and wants ₹100, betting on red at roulette (p = 18/37). With stakes of s rupees it is the same game with i = 50/s and N = 100/s. Fewer, bigger bets give the house edge fewer chances to work.

Another first-step problem: a race to roll a six

Asha and Bhavna take turns rolling a die, Asha first. The first to roll a six wins the last samosa. Conditioning on the first round: p = 1/6 + (25/36) p, so p = 6/11.

Asha also wins on a five (a tutorial exercise)

Races

Asha wins

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exact 6/11 = 0.5455

Average number of rolls

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What to notice

In a fair game (p = 1/2) the chance of reaching N is simply i/N, your share of the money on the table, and the average length is i(N − i).
Choose Ravi at roulette: a disadvantage of 1.4 percentage points per spin leaves him about a 6% chance of doubling ₹50 one rupee at a time, after about 1,600 spins on average.
Choose Deuce: tennis from deuce is this game with i = 2, N = 4. Winning 55% of points gives about a 60% chance of winning the game.
All the formulas come from one idea: condition on the first round, and find the original problem inside itself.