Probability problems

Probability problems involve calculations or reasoning that determine how likely an event is to occur. These problems rely on the principles of probability theory, which is a branch of mathematics that deals with the analysis of random phenomena. Probability can be expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event.

The "Balls into Bins" problem is a classic problem in probability and computer science, dealing with the distribution of indistinguishable balls into distinguishable bins. It can be formally defined in several ways, but a common formulation is as follows: **Problem Definition:** 1. You have \( n \) indistinguishable balls and \( m \) distinguishable bins. 2. You want to randomly place each ball into one of the \( m \) bins.

Gambler's ruin is a concept from probability theory and statistics that models a gambling scenario where a gambler continues to gamble until either they lose all their money or reach a predetermined target amount. It is often used to illustrate the principles of random walks and the behavior of stochastic processes. In a typical setup, a gambler starts with a certain amount of capital and bets on a game with a fixed probability of winning or losing.

The Pill Puzzle is a logical reasoning problem often presented as a brain teaser or puzzle. It typically involves a scenario where you have a certain number of pills, some of which are good (safe to take) and some of which are bad (harmful or lethal). The challenge often centers around identifying the good pills from the bad ones using a limited number of tests or a specific set of rules. Here's a common formulation of the Pill Puzzle: - You have a number of pills, say 12.

The "Problem of Points" is a historical problem in probability theory that deals with the question of how to fairly divide the stakes in a game when it is interrupted before the conclusion. The problem is often framed in the context of two players who are playing a game of chance, such as flipping a coin or rolling dice, and one player is ahead but the game is cut short due to an unforeseen circumstance.