Probability of Reaching a Chip Target Under Bold Betting

You start with $A$ chips and want to reach $B$ chips, where $A$ and $B$ are integers with $B > A > 0$. Each round you place an even-money bet that you win with probability $p$: a win adds the amount you staked, and a loss removes it. You play boldly: holding $x$ chips, you stake whichever is smaller, all $x$ chips or the $B - x$ chips you still need. Play stops when you reach $B$ chips or run out. What is the probability that you reach $B$ chips before going broke?

The specific values of $A$, $B$ and $p$ are generated randomly, and each question states the required output format.

Constraints and Clarifications

• $A$ and $B$ are integers with $B > A > 0$, and $p$ is a probability; write $q = 1 - p$.
• Bets pay even money, and rounds are independent.
• With $x$ chips the stake is $\min(x, B - x)$, so every stake is a whole number of chips.

Clarifying Questions

• Does a winning bet pay exactly the stake, or some other multiple of it?
• When you hold exactly half the target, is the stake all of your chips, which is also exactly what you need?
• Is the answer wanted as an expression in $p$, or as a number for the given values?