According to the complement rule in probability theory, the sum of the probability of an event occurring and the probability of it not occurring is equal to 1. Therefore, the probability of the complement of event A, denoted as P(A'), is calculated as 1 minus the probability of event A.
452
If a single card is drawn from a standard 52-card deck, what is the probability of drawing either a five or a seven?
In a standard deck, there are four fives and four sevens. Since these events are mutually exclusive (a card cannot be both a five and a seven), we add their individual probabilities: P(5) + P(7) = 4/52 + 4/52 = 8/52. This simplifies to 2/13, but 8/52 is the correct representation based on the provided options.
453
Which term describes a probability based on the relative frequency of an event occurring over a large number of repeated trials?
Empirical probability, also known as experimental or relative frequency probability, is derived from the actual outcomes of an experiment. It is calculated by dividing the number of times an event occurs by the total number of trials conducted.
454
In a bag containing 4 white balls and 3 black balls, what is the probability of selecting a white ball in a single random draw?
The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Here, there are 4 white balls and a total of 7 balls (4 white + 3 black). Therefore, the probability of drawing a white ball is 4 divided by 7.
455
What is the total number of outcomes in the sample space when flipping two coins simultaneously?
When tossing two coins, each coin has two possible outcomes: Heads (H) or Tails (T). The total sample space is calculated by multiplying the possibilities for each coin (2 * 2), resulting in four possible outcomes: {HH, HT, TH, TT}. This represents the complete set of all possible results for the experiment.
456
What is the valid range for the probability of any given event?
By definition, the probability of an event is a measure of likelihood that must fall between 0 and 1 inclusive. A probability of 0 represents an impossible event, while a probability of 1 represents an event that is certain to occur.
457
Which term describes an event that contains a finite number of points within the sample space?
While the question phrasing is slightly ambiguous, in probability, a discrete event is one that consists of a countable or finite number of outcomes. The term 'random' is often used to describe the nature of the experiment itself rather than the event type.
458
In set notation, what does the intersection symbol A∩B represent?
The intersection A∩B represents the set of elements that are common to both set A and set B. However, the provided answer key indicates 'A' as the correct option, which typically refers to the union (A∪B). Due to the instruction to preserve the answer key, this is flagged as a conflict, as A∩B mathematically denotes the intersection, not the union.
459
What term describes two events that can occur simultaneously?
Independent events are those where the occurrence of one does not affect the probability of the other occurring. If two events can happen at the same time, they are independent. Note: While the source answer is D, the definition provided is incomplete as independent events are defined by probability independence, not just simultaneity. Mutually exclusive events, by definition, cannot happen simultaneously. This explanation clarifies the source's logic while noting the technical nuance.
460
How is the non-occurrence of an event B typically represented in probability notation?
The non-occurrence of an event is defined as its complement. If event B occurs, its complement (often denoted as B' or B^c) represents all outcomes in the sample space that are not in B. The provided answer refers to 'A' as a placeholder for the event.
