Questions about Probability of Random Events .

I guess there's another sentence in front of me.

There are n possible outcomes for all possible outcomes, and if an event A contains m outcomes, then the probability of event A is p(a) = m n".

Is that so?

If so, the sentence "A contains m outcomes for a certain event" is easy to understand.

Hehe, let's give you an example.

Suppose there are 60 students in your class, and the student numbers are 1 60 in order, and one student number is randomly selected from these 60 student numbers, and the number of possible numbers is also any one of 1 60, that is, "there are 60 of all possible outcomes".

Suppose that there are 20 boys out of 60 students in your class, and "the randomly drawn student number corresponds to the boy" as event A, then, all 20 student numbers may be drawn, and the probability is the same, then event A contains 20 results (i.e., any of the 20 is drawn).

Then the probability of event a occurring is 20 60(i.e., the probability that the drawn student number corresponds to a male student).

Do you understand?? Take your time, I often encounter the phenomenon of obstruction in learning, don't be busy. Maybe in a few hours, or a few days, you will suddenly see it again.

For example, if there are 20 apples, 5 are green and the others are red. Take 1 apple at random.

A: Take out the green one.

A event contains 5 basic events, and the total number of basic events is 20.

Then the probability of a occurring is p(a)=5 20 instead of 1 5

1. Basic concepts:

1) Inevitable event: Under condition S, the event that must occur is called the inevitable event relative to condition S;

2) Impossible event: An event that must not occur under condition s is called an impossible event relative to condition s;

3) Deterministic event: Necessary event and impossible event are collectively referred to as definite event with respect to condition s;

4) Random event: an event that may or may not occur under condition S, which is called a random event relative to condition S;

5) Frequency and frequency: Repeat the test n times under the same conditions to observe whether a certain event A appears, and call event A appears in the n tests. The number of times na is the frequency of occurrence of event a; The proportion of event A occurring fn(a) = is the probability of event A occurring:

For a given random event a, if the frequency of event a fn(a) stabilizes at a certain constant as the number of trials increases, denote this constant as p(a), which is called the probability of event a.

6) The difference and connection between frequency and probability: the frequency of random events refers to the ratio of the number of times the event occurs na to the total number of experiments n, which has a certain stability, always swings around a certain constant, and with the increasing number of tests, the swing amplitude is getting smaller and smaller. We call this constant the probability of a random event, and the probability quantitatively reflects the probability of a random event occurring.

The frequency can be approximated as the probability of this event under the premise of a large number of repeated trials.

The probabilities of the occurrence of the three events are p(a), p(b), and p(c), and p=1-p(a)p(b)p(c).

Analysis: No more than two occurrences represent three kinds of emotional burial:

1.None of them happen 2Occurs 1 branch grip 3Occurs 2 pcs.

And there are four situations in total of the three events:

None of them happen 21 3Occurs 2 4All happen.

So p (no more than two events occur) = 1-p (both occurrences) = 1-p(a)p(b)p(c).

Hello, the beauty of mathematics has you covered.

2) The third time to obtain defective products, indicating that the first two times are **, which is the same as the first question.

3) The third time you get **, it means that it doesn't matter what you take the first two times.

There are three cases, 1 positive, 2 positive, 3 times, :(15 20) * (14 19) * (5 18) = 35 228

1 positive, 2 times, 3 times, :(15 20)*(5 19)*(4 18)=5 114

1 time, 2 positive, 3 times :(5 20)*(15 19)*(4 18)=5 114

The probability of getting a defective product for the third time is: (35 228) + (5 114) + (5 114) = 55 228

Random phenomenonRandomized trialsSample spaceSample pointsRandom eventsRandom variables

First of all, it is clear that the object of probability theory and mathematical statistics is:Random phenomenon。withDetermine the phenomenonCorrespondingly:

The process of repeated observation of random phenomena under the same conditions is calledRandomized trialsAt the same time, it should be noted that many random phenomena (due to conditions that cannot be perfectly reproduced) cannot be repeated, and the former is the main object of our research here.

The set of all possible fundamental outcomes of a random phenomenon is called the sample space and is denoted as , where denotes the representative element in the fundamental outcome and is calledSample points。The sample point is the most basic unit of sampling.

Part of a random phenomenon.

The set of sample points is calledRandom events, abbreviationEvents, which is generally indicated by capital letters. Events can be described in precise language or represented as collections.

An event is a subset of the corresponding sample space. A common tool for topological properties in probability theory is:Venn diagram, usually with a large rectangle representing the entire sample space, and an arbitrary enclosed figure (usually a circle) inside the rectangle representing the event.

When a sample point in a subset occurs, an event is said to have occurred; Another way to put it is that an event occurs when and only when a sample point occurs.

A subset of individual elements in a sample space is calledBasic events

The largest subset of the sample space ( itself ) is calledInevitable event

The smallest subset of the sample space ( ) is calledImpossible event

The variable that represents the outcome of a random phenomenon is calledRandom variables, generally represented by uppercase Latin letters.

Random events

The easiest way to toss a coin: toss a coin, the result may be face up, or face up, put a table tennis ball of different colors in the box, take one with your eyes closed, the color of the ball is randomly burning, the candle is blown by the wind, the 26 letters are shuffled and respelled, the dice are thrown, what will be the weather tomorrow, etc.

Determine the event: The sum of the two sides of the triangle is greater than the third side.

The sun must have risen in the east.

Parallel lines do not intersect.

No two leaves are the same in the world.

A triangle is made up of 3 sides.