Events are described as 'equally likely' if there is no reason to expect one event to occur more frequently than another. In a fair experiment, such as rolling a balanced die, each individual outcome has an equal probability of 1/6.
482
In set theory and probability, how is the joint event of two events G and H represented?
A joint event occurs when two or more events happen simultaneously. In probability notation, the joint occurrence of events G and H is represented by the intersection of the two sets, denoted as G ∩ H. This signifies the set of outcomes that are common to both event G and event H.
483
In probability theory, if A and B are two events, how is the event representing the occurrence of A, B, or both denoted?
The union of two events, denoted as A ∪ B, represents the set of all outcomes that are in A, in B, or in both. It captures the logical 'OR' condition in probability, encompassing all possibilities where at least one of the defined events occurs.
484
If the occurrence of one event precludes the possibility of another event happening simultaneously, what term describes these events?
Events are defined as mutually exclusive if they cannot occur at the same time. In set theory, this means the intersection of the two events is the empty set. For example, when flipping a single coin, the outcomes 'Heads' and 'Tails' are mutually exclusive because the coin cannot land on both sides simultaneously. This is a fundamental concept in probability theory used to simplify the calculation of the probability of the union of events.
485
Given four machines with independent failure probabilities of 0.24, 0.45, 0.35, and 0.38, what is the probability that all four machines fail before the next year?
Assuming the failure events are independent, the probability of all machines failing is the product of their individual probabilities: 0.24 * 0.45 * 0.35 * 0.38 = 0.014364. However, based on the provided answer key, there appears to be a discrepancy between the calculated result and the provided option. The calculation 0.24 * 0.45 * 0.35 * 0.38 equals approximately 0.0144, which does not match 0.168.
486
How are two events classified if their occurrence is linked or dependent on one another?
In probability theory, a compound event is an event that is formed by combining two or more simple events. If the occurrence of one event affects the probability of the other, they are considered dependent, and their joint occurrence is a compound event.
487
In the context of a random experiment, what term is used to describe the individual observations or outcomes of a random variable?
A trial is a single performance of a random experiment. Each trial results in an outcome. When we observe a random variable across multiple repetitions of an experiment, each repetition is referred to as a trial, which contributes to the overall set of data points.
488
What is the probability of selecting a diamond card from a standard deck of 52 playing cards?
A standard deck contains 52 cards divided into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards. Therefore, the probability of drawing a diamond is the number of diamond cards (13) divided by the total number of cards (52), which simplifies to 1/4.
489
When flipping a fair coin, what is the probability of obtaining a head?
A fair coin has two possible outcomes: heads or tails. Since each outcome is equally likely, the probability of getting heads is 1 out of 2, which is 0.5 or 50%. This represents the total probability space for a single coin flip.
490
What is the probability of obtaining an odd sum when rolling two fair dice?
When rolling two dice, there are 6 * 6 = 36 possible outcomes. A sum is odd if one die is even and the other is odd, or vice versa. There are 18 such combinations (e.g., 1+2, 1+4, 1+6, 2+1, 2+3, 2+5, etc.). Since 18 out of 36 outcomes result in an odd sum, the probability is 18/36, which simplifies to 1/2.
