Intereting Posts

What is the definition of a set?
a q-continued fraction related to the octahedral group
Why does L'Hopital's rule fail in this case?
If $f$ is nonnegative and continuous on $$, then $\left(\int_a^b f(x)^n \ dx\right)^{1/n}\to\max\limits_{} f$
idempotents acting as local identities
Proof of uniform convergence and continuity
What is limit of $\sum \limits_{n=0}^{\infty}\frac{1}{(2n)!} $?
Show bounded and convex function on $\mathbb R$ is constant
how to solve $\sum _{m=0}^{k-1}mC_{k-1}^{m}C_{N-k}^{m}$?
Size of a family of sets $F$ such that if $|A\cap X|=|B\cap X|$ for all $X\in F$, then $A=B$
How to either prove or disprove if it is possible to arrange a series of numbers such the sum of any two adjacent number adds up to a prime number
Let $K$ be a field and $f(x)\in K$. Prove that $K/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K$.
How to find $\zeta(0)=\frac{-1}{2}$ by definition?
Induction Proof: Formula for Sum of n Fibonacci Numbers
Proving the integral of an inverse function

In each round, the gambler either wins and earns 1 dollar, or loses 1 dollar. The winning probability in each round is $p<1/2$. The gambler initially has $a$ dollars. He quits the game when he has no money, or he has lost $k>a$ rounds in all by this time, no matter how many rounds he wins. (For example, if $a=2$, $k=3$, and the sequence is +1,+1,+1,-1,+1,-1,-1, he quits now.) What is his expected exit time?

What confuses me is the dependence between these two events. I know the generating function of the exit time in the standard gambler’s ruin problem, and the duration until the gambler loses $k$ dollars in all is a negative binomial random variable. But these two stopping times are dependent. I was wondering if anyone could give me some hint. Thanks a lot!

**Update:** From Ross Millikan’s hint: how to calculate the probability that the wealth is $b$ at the end of round $2k-a$, given that the game is not over?

- Eigenvalues for $3\times 3$ stochastic matrices
- How can a Markov chain be written as a measure-preserving dynamic system
- Flea on a triangle
- Generalization of the Jordan form for infinite matrices
- Stopping time in Markov chains
- Conditional return time of simple random walk

- Expectation of ratio of sums of i.i.d. random variables. What's wrong with the simple answer?
- Finding $\mathbb{E}$, $X \sim \text{Pareto}$ under exam conditions
- Randomly dropping needles in a circle?
- Overlapping Probability in Minesweeper
- Combinatorics: Number of possible 10-card hands from superdeck (10 times 52 cards)
- Error propagation, why use variences?
- Combinatorics with simple substitution ciphers question
- Gaussian integral evaluation
- Average number of flips for someone to win?
- Quick way to tell if a set of dice is NOT non-transitive

For the loss of $k$ to kick in, he needs to win $k-a$ times. If he does that, he will never go broke (except maybe on the round he would quit because of the $k$ losses). He needs to win those $k-a$ within the first $2k-a$ games. So compute the chance he goes broke in less than $2k-a$ games and the expected length of a game in that scenario. This gives you the chance he invokes the $k$ losses. Now compute the expected length of a game given that he wins at least $k-a$ in the first $2k-a$

- Limits defined for negative factorials (i.e. $(-n)!,\space n\in\mathbb{N}$)
- How can I find the square root using pen and paper?
- Completeness of Metric Space and Measure Space
- Chebyshev function identity
- Why is $(1+\frac{3}{n})^{-1}=(1-\frac{3}{n}+\frac{9}{n^2}+o(\frac{1}{n^2}))$ and how to get around the Taylor expansion?
- Hyperbolic area and $SL_2$
- How to solve Diophantine equations of the form $Axy + Bx + Cy + D = N$?
- Can anybody explain about real linear space and complex linear space?
- Limit of an expression
- Use Gröbner bases to count the $3$-edge colorings of planar cubic graphs…
- How do you calculate this limit without using L'Hopital or Taylor?
- Homology of the Klein Bottle
- Entropy of matrix
- Is symmetry a valid option in inequalities?
- On average, how many friends would I need to have to have at least one friend's birthday every day?