Final Answer:
The correct explanation is: 1) The area under the curve represents the probability of an event occurring.
Step-by-step explanation:
In probability theory, the area under a probability density function (PDF) curve within a certain interval corresponds to the probability of an event occurring within that interval. The total area under the curve of a probability distribution function is always equal to 1, representing the total probability of all possible outcomes. Probability density functions are utilized to model continuous probability distributions, and the area under such curves provides insights into the likelihood of different events or values occurring within a specified range. Therefore, option 1 accurately describes the significance of the area under a curve in terms of probability.
When dealing with probability distributions, the integral of the PDF over a given interval yields the probability of the associated event occurring within that interval. The integral effectively calculates the area under the curve, and for a valid probability distribution function, this area must always sum up to 1. This property ensures that the total probability across all possible outcomes within the distribution is accounted for. Thus, the area under the curve serves as a representation of probabilities and is a fundamental concept in probability theory.
Understanding the relationship between the area under a probability distribution curve and probabilities is crucial in various fields, including statistics, data analysis, and risk assessment. It provides a quantitative measure of the likelihood of certain events or values occurring within a continuous distribution, aiding in decision-making and drawing inferences based on probability assessments. Therefore, recognizing that the area under the curve signifies the probability of events occurring is a fundamental concept in probability theory and its practical applications.