The gambler's ruin problem is a classic application of conditional probability and first-step analysis: given a random walk (or equivalently, two players repeatedly exchanging a unit stake) with absorbing boundaries at 0 and N, first-step analysis conditions on the outcome of the first step to derive a linear difference equation relating the win-probability at state i to the win-probabilities at states i+1 and i−1, subject to boundary conditions at the absorbing states. Difference equations of this form are solved via a guess-and-verify method (trying a geometric form, reducing to a characteristic quadratic, and combining roots into a general solution fit to boundary conditions), yielding a closed-form win probability and, as a corollary, proof that the process terminates with probability one. The lecture also introduces the random variable as a formal concept: a measurable function from a sample space to the real line that assigns a numerical summary to an aspect of a random experiment, with randomness originating in the underlying experiment rather than in the function itself, illustrated via the Bernoulli and binomial distributions as foundational named distributions.
The gambler's ruin problem is a classic application of conditional probability and first-step analysis: given a random walk (or equivalently, two players repeatedly exchanging a unit stake) with abso…