If the sum of the probabilities of two events is 1, are these two events opposites?

Not really, or not necessarily. TwoOpposing eventsThe sum of the probabilities is 1, and the reverse is not true. For example, the probabilities of "Toss a coin to get heads" and "Today's 3d single digits are singular" are both 1, but they are not opposites.

For example, "rabbits are animals" is true, but "animals are rabbits" is clearly wrong.

Probability, also known as "probability", is a reflection of random events.

The probability of occurrence. A random event is an event that may or may not occur under the same conditions. For example, from a batch of ** and defective goods, randomly draw a piece, "draw **" is a random event.

Suppose n experiments and observations are carried out on a random phenomenon, in which event a occurs m times, that is, the frequency of its occurrence is m n. After a lot of trial and error, it is common for m n to get closer and closer to a certain constant (see Bernoulli's law of large numbers for this proof). This constant is the probability of event a occurring, which is often denoted by p (a).

History: The first person to systematically calculate probabilities was Cardano in the 16th century. It is recorded in his book "Liber de Ludo Aleae". The content of the book about probability was written by Gould from Latin.

Translated.

Cardano's mathematical writings have a lot of advice for gamblers. These suggestions are written in short essays. However, it was Pascal who first proposed a systematic study of probability.

In a series of correspondence with Fermat. These communications were originally made by Pascal, who wanted to ask Fermat a few questions about the questions posed by Chevvalier de Mere. Chevvalier de Mere was a well-known writer, Louis XIV.

The dignitary of the court was also an avid gambler. There are two main problems: the craps roll problem and the prize money distribution problem.

p(ab)=1/12。Since p(a b) = p(a) + p(b)-p(ab), then p(b) = p(a b) + p(ab)-p(a) = 1 2 + 1 4-1 3 = 5 12;p(b|a)=p(ab) p(a)=1 3 so p(ab)=1 12;p(a|b)=p(ab)/p(b)=1/2；Therefore, p(b)=1 6;p(a∪b)=p(a)+p(b)-p(ab)=1/4+1/6-1/12=1/3。

p(ab) are two independent events.

The probability of simultaneous occurrence is equal to the product of the probability of occurrence of each event, i.e., p(a b) = p(a) p(b). p(a·b), the dot product in the middle.

It is generally not omitted to indicate that it is two events, not event ab (one event).

p(a·b) indicates the probability that event a and event b occur at the same time, and the reason why this notation is used is because when studying the situation that event a and event b occur at the same time, the most common situation encountered is that a and b are unrelated or independent of each other, in this case p(a·b) = p(a)·p(b), it can be seen that this notation is very concise and easy to remember.

It should be noted that p(a·b)=p(a)·p(b) in the examination is generally not valid, that is, a, refers to the fact that b is not independent, and a general formula is often used.

Conditional Probability Formula:

p(a|b) =p(ab)/p(b)

p(a|b) – the probability of a under b condition. That is, the probability of event a occurring under the condition that another event b has already occurred.

p(ab) – the probability that events a and b occur at the same time, i.e., the joint probability. Joint probability indicates the probability that two events will occur together. The joint probability of a and b is expressed as p(ab) or p(a,b).

p(b) – the probability of event b occurring.

Conditional probability Example: This is the probability of event A occurring under the condition that another event B has already occurred. The conditional probability is expressed as p(a|b), which is read as "probability of a under condition b".

The probability of a necessary event occurring is 1, but an event with a probability of 1 is not necessarily a necessary event. The continuous random variable x is obtained by eliminating the finite points from the entire sample space if the probability of any finite number of points in the sample space is 0'It should not be a finite number of points'The probability remains at 1. (It can be understood by analogy with the existence of a finite number of removable discontinuities in a high number of integrals that does not affect the integral value.)

Necessary events are called definite events in conjunction with impossible events, so necessary events do not include impossible events.

Although the probability of a necessary event is 1, an event with probability 1 is not necessarily a necessary event.

For example, if a target is large enough, the probability of hitting a target in addition to the bullseye is 1, but it is not inevitable, because it can also hit the bull's-eye.

Hello! Not necessarily, for example, if you take a point on [0,1] on the number line, the probability of not taking it is 1, but it is not a necessary event. Similarly, an event with a probability of 0 is not necessarily an impossible event.

In the previous example, the probability of getting is 0). The Economic Mathematics team will help you solve the problem, please adopt it in time. Thank you!

Is an event with a probability of 1 necessarily an inevitable event?

A: Not necessarily.

For example, let the continuous random variable x be in a closed interval.

0,1]. Let event a be defined as:

a=--- note that it is an open interval and does not include 0 and 1.

p(a)=1.

But x=0 or x=1 is possible. That is, a does not necessarily happen.

In classical generalizations, this sentence is not true. Because there are finite elements in the sample space, "impossible events" and "events with zero probability" are equivalent, as are "inevitable events" and "events with one probability".

In geometric generalizations, this is true. Let me start with an example to illustrate that on the interval [0,1], the probability of getting a point is zero, but" taking this event is possible, not an "impossible event".

This is because there are an infinite number of elements in the sample space in the geometric generalization, and the measurement of the scale of the geometric region requires the help of metric theory, and we know that the measure of the closed interval on a straight line is the usual length of the line segment. And a single point, the measure is 0, so there is a zero probability.

It's not just a point, it's that the measure of the whole rational number is 0, although that sounds hard to accept. So the probability of "getting a rational number" in the interval [0,1] is also zero.

On the interval [0,1], the measure of all irrational numbers is 1, so the probability of "taking irrational numbers" is 1, which is obviously not a necessary event, because I may also take rational numbers.

Summary. Not necessarily, the two events are not necessarily mutually exclusive. The sum of the probabilities of the occurrence of the two events is equal to the sum of the probabilities of the occurrence of the two events, which is true only when the two events are mutually exclusive.

Workaround:1First of all, to clarify whether two events are mutually exclusive, you can judge by observing the actual situation, if two events can occur at the same time, they are not mutually exclusive; 2.

Secondly, it can be judged by calculating the probability, if the sum of the probabilities of the two events is greater than 1, it means that the two events are not mutually exclusive; 3.Finally, it can be judged by constructing a probability tree, if two nodes of events in the probability tree can occur at the same time, it means that the two events are not mutually exclusive.

The probability of two parallel events is equal to the sum of the probabilities of the two events, and the two events must be mutually exclusive.

Not necessarily, the two events are not necessarily mutually exclusive. The sum of the probabilities of the occurrence of two events is equal to the sum of the probabilities of the occurrence of the two events, and is only true when the two events are mutually exclusive. Workaround:

1.First of all, it is necessary to clarify whether the two events are mutually exclusive, which can be judged by observing the actual situation, and if the two events can occur at the same banquet, they are not mutually exclusive; 2.Secondly, it can be judged by calculating the probability, if the sum of the probabilities of the two events is greater than 1, it means that the two events are not mutually exclusive; 3.

Finally, it can be judged by constructing a probability tree, if two nodes in the probability tree can occur at the same time, it means that the two events are not mutually exclusive.

Excuse me, but please go into more detail?

Not necessarily. Two events are not necessarily mutually exclusive, even if the sum of the probabilities of their occurrence is equal to 1. The definition of mutual exclusion is:

It is not possible for two events to occur at the same time, i.e. the sum of their probabilities is 1. For example, if the probability of event A occurring is and the probability of event B occurring is , then the sum of the probabilities of event A and event B is also 1, but they are not mutually exclusive because they can occur at the same time. In addition, the two events may also be related, i.e. the sum of the probabilities of their occurrence is fixed at 1.

For example, if the probability of event A occurring is and the probability of event B occurring before event B is , then the sum of the probabilities of event A and event B is that event A and event B are related and that they may occur at the same time. In short, the sum of the probabilities of two events is equal to 1, which does not mean that they are mutually exclusive or correlated, but only that they can be judged by calculating the sum of their probabilities whether they are mutually exclusive or related.