A standard deck contains 52 cards. Picture cards are defined as Jacks, Queens, and Kings. Since there are 4 suits, each having 3 picture cards, the total number of picture cards is 4 * 3 = 12. Thus, the probability of drawing a picture card is 12/52.
472
What is the probability value assigned to a certain (sure) event?
In probability theory, the probability of an event ranges from 0 to 1. A certain event, which is guaranteed to occur in every trial of an experiment, is assigned a probability of 1.
473
If two events, A and B, are mutually exclusive, what is the probability of both events occurring simultaneously?
By definition, mutually exclusive events cannot occur at the same time. Therefore, the intersection of the two events is an empty set, and the probability of both A and B occurring simultaneously, denoted as P(A ∩ B), must be equal to zero. This is a foundational concept in probability theory.
474
Which of the following values cannot represent the probability of an event?
According to the fundamental axioms of probability, the probability of any event must lie within the closed interval [0, 1]. Therefore, a probability value can never be negative, nor can it exceed 1. Zero represents an impossible event, while positive values represent possible events.
475
What is the correct addition rule for the probability of the union of two partially overlapping events X and Y?
The general addition rule for probability states that for any two events X and Y, the probability of their union is the sum of their individual probabilities minus the probability of their intersection. This subtraction is necessary to avoid double-counting the outcomes that are common to both events when they overlap.
476
How is the product of the first 'n' natural numbers mathematically represented?
The product of the first n natural numbers is defined as the factorial of n, denoted as n!. This is calculated by multiplying all positive integers from 1 up to n. Therefore, the sequence is 1 * 2 * 3 * ... * (n-1) * n. Option B correctly represents this sequence of multiplication. Note that the product does not include zero, as multiplying by zero would result in a product of zero for all n.
477
Which term defines the relationship between two events where the occurrence of one event precludes the occurrence of the other?
In probability theory, events are defined as mutually exclusive if they cannot occur simultaneously. If event A happens, event B cannot happen, and vice versa. This concept is fundamental in probability theory for calculating the likelihood of combined events and is a cornerstone of statistical analysis and set theory applications.
478
In probability theory, how are events that cannot occur simultaneously described?
Mutually exclusive events are defined as events that cannot happen at the same time. If one event occurs, the probability of the other occurring is zero.
479
In a standard deck of 52 playing cards, what is the probability of drawing either a diamond or a club?
A standard deck contains 52 cards divided into four suits of 13 cards each. There are 13 diamonds and 13 clubs. Since these events are mutually exclusive, the probability of drawing a diamond or a club is the sum of their individual probabilities: 13/52 + 13/52 = 26/52, which simplifies to 1/2.
480
What is the term for an experiment that is repeated multiple times in sequence?
A random experiment is a process that can be repeated under similar conditions, where the outcome is not deterministic but belongs to a well-defined set of possible results. Repeated trials allow for the observation of long-term frequencies.
