Final answer:
The student question pertains to probability and the use of a random variable X to calculate probabilities of certain events occurring, such as the number of cracked eggs in a sample or batches sold. The value of X represents the number of occurrences and is used to find probabilities through relevant statistical methods.
Step-by-step explanation:
The question revolves around the concept of probability and the use of a random variable denoted by X. Specifically, it deals with calculating the probability of a certain number of events, such as cracked eggs or selling batches, occurring when the events follow a specific distribution. In the context of the given excerpts, X represents the number of occurrences of an event, and we are interested in finding probabilities such as P(X = x) for exact outcomes or using inequalities like P(X ≥ x) for a range of outcomes. To compute these probabilities, we may use various probability distributions like binomial, geometric or others, depending on the scenario.
In one example related to eggs, where there are 12 cracked eggs in a gross of 144, we would define X as the number of cracked eggs found in a sample of 15. We then use the binomial probability formula to calculate the probability of finding at most three cracked eggs in the sample. Similarly, when dealing with batches sold by a baker or employing the quadratic formula to find a value, understanding how to define and work with the random variable X is essential to answering such questions.