The relative frequency approach to probability defines the probability of an event as the limit of the ratio of the number of occurrences of the event to the total number of trials as the number of trials increases. This empirical method relies on observed data from experiments rather than theoretical assumptions, making it a cornerstone of frequentist probability theory.
442
In probability theory, what term describes an event whose probability is equal to one?
In the axiomatic definition of probability, the probability of the entire sample space is 1. An event that corresponds to the entire sample space is defined as a certain event, meaning it is guaranteed to occur in any given trial.
443
For two mutually exclusive events A and B, which rule allows the calculation of the probability of either event occurring by summing their individual probabilities, P(A) + P(B) = n(A)/n(S) + n(B)/n(S)?
The addition rule for probability states that for any two events, P(A or B) = P(A) + P(B) - P(A and B). When events are mutually exclusive, they cannot occur simultaneously, meaning P(A and B) = 0. Therefore, the probability of either event occurring simplifies to the sum of their individual probabilities, which is expressed as the sum of their relative frequencies.
444
In probability theory, what is the specific term for an individual outcome within a sample space?
The sample space is the set of all possible outcomes of a random experiment. Each individual element or outcome within this set is referred to as a sample point. An event is defined as a subset of the sample space.
445
What is the probability of rolling either an even or an odd number on a fair six-sided die?
A fair six-sided die has outcomes {1, 2, 3, 4, 5, 6}. There are 3 even numbers (2, 4, 6) and 3 odd numbers (1, 3, 5). Since all outcomes are either even or odd, the probability of rolling an even or odd number is 6/6 = 1. The option 3/6 is mathematically equivalent to 1/2, though the event is certain.
446
Under what condition is the special rule of multiplication of probability applicable?
The special rule of multiplication states that for two events A and B, P(A and B) = P(A) * P(B). This rule is valid if and only if the events are independent, meaning the occurrence of one event does not affect the probability of the occurrence of the other.
447
In the context of probability experiments, how are the possible outcomes formally defined?
In probability theory, an experiment is a process that leads to one of several possible outcomes. A subset of the sample space, which represents the set of all possible outcomes of an experiment, is referred to as an event. Events are the fundamental units used to assign probabilities.
448
Given two events Y and Z, how is the intersection of the occurrence of Z and the non-occurrence of Y denoted?
The non-occurrence of event Y is represented by the complement of Y, denoted as Y-bar (or Y'). The intersection of this complement with event Z represents the set of outcomes where Z occurs and Y does not occur, which is mathematically expressed as Y-bar ∩ Z.
449
What is the collective term for the set of all possible outcomes resulting from a random experiment?
The sample space is the set of all possible outcomes of a random experiment. In probability theory, an event is a subset of the sample space. Because the sample space itself is a set of outcomes, it is sometimes referred to as the event space in specific contexts, making both terms applicable depending on the theoretical framework used.
450
In a container holding 12 balls, where 6 are red and 6 are white, what is the probability of randomly selecting a white ball?
The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Here, there are 6 white balls out of a total of 12 balls. Therefore, the probability is 6/12, which simplifies to 0.5 or 50%.
