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Opposing events. Refers to two mutually exclusive events in which one of them must occur.
This is the theory of probability.
The term, also known as "inverse events", cannot occur at the same time.
If A and B are impossible events and A and B are inevitable events, then event A and event B are opposites of each other, which means that event A and event B must have one and only one of them. That is, in each trial, one of the events A and B must occur, and only one of them occurs.
The relationship between the probabilities of opposing events: p(a) + p(b) = 1. For example, in the craps rolling test, a=,b=,a b is an impossible event and a b is a necessary event, so a and b are opposites of each other.
Mutual exclusion and opposites.
1. Different angles are targeted. The former is about whether it can occur at the same time, that is, two mutually exclusive events mean that they cannot occur at the same time; The latter is aimed at events that have no effect, i.e. two independent of each other.
Refers to the fact that the occurrence of one event has no effect on the probability of another event occurring (note: it is not that the occurrence of one event has no effect on the occurrence of another event).
2. The number of tests is different. The former is a different event that occurs under one trial, and the latter is a different event that occurs under two or more different trials.
3. The probability formula is different. If a and b are mutually exclusive events, then there is a probability addition formula p(a+b)=p(a)+p(b), and if a and b are not mutually exclusive events, then there is the formula p(a+b)=p(a)+p(b)-p(ab); If a and b are independent events, then there is a probability multiplication formula p(ab) = p(a)p(b).
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If A and B are impossible events and A and B are inevitable events, then Event A and Event B are said to be opposites of each other.
In layman's terms, A and B cannot happen at the same time, but one of them will happen, and they are a life-and-death relationship.
Unlike mutex events, mutex events cannot occur at the same time, but it is possible that they do not occur at all.
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Difference: Opposites must inevitably accept each other's thoughts, and mutual rejection is not necessarily opposite. The popular understanding of the opposing event: A does not happen, B definitely happens.
The popular understanding of mutually exclusive time: A and B cannot occur at the same time.
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An opposing event is one of the two mutually exclusive events in which one of them must occur. This is a probabilistic term. Also known as "inverse events", it is impossible for them to happen at the same time.
If A and B are opposites, then A and B are mutually exclusive, and/or are inevitable, i.e., in an experiment, Event A and its opposing event B can only occur in one, and one must occur in a group, and it is impossible for neither to occur or both.
Let's take a look at the example questions.
Example 1(True/False).
1.In the same experiment, it is correct to assume that A and B are two random events, and that "if A and B are two opposing events". (×
Answer] Analysis] Obviously, this statement is false The opposing event is a special case of mutually exclusive events, and in addition to satisfying a b, a b must also be a necessary event.
For example, if you toss a coin, the result can be heads or tails, but the third outcome will not appear.
Focus: The relationship between opposing events and mutually exclusive events.
If event A and event B are opposites, then event A and event B must be mutually exclusive.
Conversely, if event A and event B are mutually exclusive events, then event A and event B are not necessarily opposites.
Example 1(True/False) Mutually exclusive events are not necessarily opposites. (
Answer] 1√。
Analysis] Opposing events are special mutually exclusive events.
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Use <> to indicate the occurrence of no more than two of the three random events.
Analysis: no more than two occurrences, that is, including one occurrence and two occurrences; That is to say, it is an opposing event of three events at the same time.
The simultaneous occurrence of three events can be expressed as:
According to the formula for calculating the probability of opposing events: p(a)+p(b)=1. then the three events do not occur at the same time, that is, no more than two occurrences can be expressed as:
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Difference: "Opposing events" and "mutually exclusive events" have an inclusive relationship, the number of events in "mutually exclusive events" can be two or more, while "opposing events" are only friendly for two events, and the opposition of two events is a sufficient condition for the mutual exclusion of these two events, but it is not a necessary condition.
An antagonistic event is a special kind of mutually exclusive event. There are two special points: first, the number of events is special (only two events); Second, the occurrence is special (there is only one occurrence).
If A and B are opposites, then A and B are mutually exclusive and A+B is a necessary event, so the probability of A+B occurring is 1, that is, P(A+B) = P(A) + P(B) = 1.
Opposites are necessarily mutually exclusive, and mutually exclusive are not necessarily opposites.
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"At least one of the opposing events that occurred" is an event that does not occur at all in the ABC. An opposing event is two mutually exclusive events in which one of them must occur, which is a probabilistic term, also known as an "inverse event", which cannot occur at the same time.
If A and B are impossible events and A and B are inevitable events, then event A and event B are mutually opposed events, which means that event A and event B must have one and only one of them, and the connection between the mutually exclusive event and the opposing event is that the opposing event is a special kind of mutually exclusive event.
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Two opposing events mean that only one of the two events occurs in each trial. For example, for a coin toss test, a="tails on top" and b="heads on top", it is clear that a and b are opposite events.
The exact mathematical description is: a, b are opposite events, p(ab)=0, p(a)+p(b)=1
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