The passage presents a significant portion of Thomas Bayes' famous essay, "An Essay Towards Solving a Problem in the Doctrine of Chances." The essay addresses the problem of determining the probability of an unknown event occurring within a given range, based on the number of times it has happened and failed. The essay is divided into sections, with definitions and propositions that form the foundation of the analysis. **Section 1** introduces key definitions: 1. Inconsistent events are those where if one occurs, none of the others can. 2. Contrary events are such that one or the other must occur, but both cannot. 3. An event is said to fail when its contrary has occurred. 4. An event is determined when it has either happened or failed. 5. The probability of an event is the ratio of the expected value to the actual value. **Proposition 1** states that if several events are inconsistent, the probability of one or the other occurring is the sum of their individual probabilities. This is demonstrated through an example involving three events, each with a specific probability, and the calculation of the total expected value. **Corollary** to Proposition 1 explains that if it is certain that one of three events must occur, the sum of their probabilities must equal 1. This leads to the understanding that the probability of an event plus the probability of its failure is the ratio of equality. **Proposition 2** discusses the relationship between the probability of an event and the gain or loss associated with it. It states that the probability of an event is to the probability of its failure as the loss if the event fails to the gain if the event occurs. This proposition is illustrated with an example involving an expectation of receiving a certain amount based on the occurrence of an event.The passage presents a significant portion of Thomas Bayes' famous essay, "An Essay Towards Solving a Problem in the Doctrine of Chances." The essay addresses the problem of determining the probability of an unknown event occurring within a given range, based on the number of times it has happened and failed. The essay is divided into sections, with definitions and propositions that form the foundation of the analysis. **Section 1** introduces key definitions: 1. Inconsistent events are those where if one occurs, none of the others can. 2. Contrary events are such that one or the other must occur, but both cannot. 3. An event is said to fail when its contrary has occurred. 4. An event is determined when it has either happened or failed. 5. The probability of an event is the ratio of the expected value to the actual value. **Proposition 1** states that if several events are inconsistent, the probability of one or the other occurring is the sum of their individual probabilities. This is demonstrated through an example involving three events, each with a specific probability, and the calculation of the total expected value. **Corollary** to Proposition 1 explains that if it is certain that one of three events must occur, the sum of their probabilities must equal 1. This leads to the understanding that the probability of an event plus the probability of its failure is the ratio of equality. **Proposition 2** discusses the relationship between the probability of an event and the gain or loss associated with it. It states that the probability of an event is to the probability of its failure as the loss if the event fails to the gain if the event occurs. This proposition is illustrated with an example involving an expectation of receiving a certain amount based on the occurrence of an event.