No, probability values cannot be negative or greater than 1. The probability of an event is a measure of the likelihood of the event occurring and is always expressed as a number between 0 and 1, inclusive. A probability of 0 indicates that the event will not occur, while a probability of 1 indicates that the event will occur. Any value outside this range is not valid in the context of probability. This is a fundamental principle in probability theory and ensures that probabilities are coherent and applicable in real-world contexts, providing a valid measure of uncertainty.

Mutually exclusive events are events that cannot occur simultaneously. In the context of probability, if two events A and B are mutually exclusive, the occurrence of A eliminates the possibility of B occurring, and vice versa. Mathematically, this is expressed as P(A and B) = 0. When calculating the probability of either of two mutually exclusive events occurring, i.e., P(A or B), you simply add their individual probabilities together: P(A or B) = P(A) + P(B). This is known as the Addition Rule for Mutually Exclusive Events. It’s crucial to identify mutually exclusive events accurately to ensure correct probability calculations in various applications.

A probability distribution provides a comprehensive overview of the likelihood of all possible outcomes of a random variable. It assigns a probability to each outcome in the sample space, ensuring that each probability is non-negative and that the sum of all the probabilities is equal to 1. The sample space and the probability distribution are intrinsically linked. The sample space provides all the possible outcomes, while the probability distribution assigns a probability to each of these outcomes, ensuring a systematic and structured approach to analysing random phenomena. This allows statisticians and data scientists to predict, model, and analyse random processes and events effectively, providing a foundation for inferential statistics and hypothesis testing.

Theoretical probability is derived based on the possible outcomes in the sample space, assuming each outcome is equally likely without conducting any experiment. It is calculated as the ratio of the number of favourable outcomes to the total number of possible outcomes. On the other hand, experimental or empirical probability is based on actual experiments and is calculated as the ratio of the number of times an event occurs to the total number of trials conducted. While theoretical probability is based on inherent likelihood, experimental probability is derived from actual data and observations, and the two may not always be the same due to the randomness and variability in real-world experiments.

Probability plays a crucial role in real-life decision making across various fields such as finance, medicine, and engineering. For instance, in finance, investors often use probability to assess the risk of a particular investment outcome. They analyse historical data to determine the likelihood of future returns. In medicine, probability is used to predict the effectiveness of a treatment or the likelihood of a particular side effect occurring. Engineers may use probability to evaluate the reliability of systems and to predict failure rates. Essentially, understanding probability allows individuals and professionals to make informed decisions by assessing the likelihood of different outcomes and thereby managing risk effectively.