The earliest known applications of probability theory in India were in the context of gambling, particularly in games involving dice.

Brahmagupta, a renowned Indian mathematician and astronomer, is widely recognized for his contributions to probability theory, including his work on calculating probabilities in dice games.

The Lilavati, written by Bhaskara II, is a significant mathematical treatise that includes a section dedicated to probability and chance, known as the 'Ganita-adhyaya'.

Bhaskara II's problem in the Lilavati focuses on calculating the probability of obtaining a specific sum when rolling two dice, laying the foundation for the study of probability distributions.

Bhaskara II introduced the concept of 'mathematical expectation' in probability theory, which is the average value of a random variable over all possible outcomes.

The Indian contributions to probability theory were instrumental in laying the groundwork for the development of modern probability theory, providing insights into the behavior of random phenomena, and influencing the development of statistical methods.

Probability theory was applied in ancient India not only in gambling but also in astronomy, medicine, and agriculture, demonstrating its versatility and wide-ranging applications.

Indian mathematicians combined both empirical observations and experimentation with deductive reasoning and mathematical proofs to study probability, demonstrating a holistic approach to the subject.

Bhaskara II is renowned for his work on the problem of points, which involves determining the probability of winning a game of chance based on the number of points scored by each player.

The problem of points holds significance in the history of probability theory as it laid the foundation for the development of modern probability theory, provided a framework for analyzing games of chance, and influenced the development of statistical methods.

Indian mathematicians made significant contributions to the development of statistical methods by developing methods for collecting and analyzing data, introducing the concept of sampling, and devising techniques for estimating population parameters.

Bhaskara II developed a method for calculating the area of a triangle using probability, demonstrating the versatility and applicability of probability theory beyond games of chance.

Indian mathematicians employed probability theory to solve problems in astronomy, including predicting the positions of celestial bodies, estimating the distances to stars and planets, and calculating the probability of eclipses.

Bhaskara II is renowned for his work on the problem of the gambler's ruin, which involves determining the probability of a gambler eventually losing all their money in a game of chance.

The problem of the gambler's ruin holds significance in the history of probability theory as it provided insights into the behavior of random processes, influenced the development of modern probability theory, and laid the foundation for the study of risk and ruin in gambling.