1, basic concept:

(1) inevitable event: the event that will happen under condition S is called the inevitable event relative to condition S;

(2) Impossible events: events that will not happen under condition S are called impossible events relative to condition S;

(3) Deterministic events: inevitable events and impossible events are collectively referred to as deterministic events relative to condition S;

(4) Random events: events that may or may not occur under condition S are called random events relative to condition S;

(5) Frequency and number of times: repeat the test for n times under the same condition S, and observe whether there is an event A, and the number of times that the event A appears in the n tests is called event nA.

Frequency of occurrence of segment a; The ratio fn(A)=n when event a occurs.

Is the probability of event A: for a given random event A, if the frequency fn(A) of event A is stable at a certain constant with the increase of test times, this constant is recorded as P(A), which is called the probability of event A ...

(6) Difference and connection between frequency and probability: The frequency of a random event refers to the ratio of the number of times nA of the event to the total number of times n of experiments, which has certain stability and always swings around a certain constant, and the swing amplitude becomes smaller and smaller with the increase of the number of experiments. We call this constant the probability of random events, which quantitatively reflects the probability of random events. Frequency can be approximated as the probability of the event under the premise of a large number of repeated experiments.

Basic properties of probability

1, basic concept:

The inclusion, union, intersection and equality of (1) events.

(2) If A∩B is an impossible event, that is, A ∩ B = Ф, then event A and event B are mutually exclusive;

(3) If A∩B is an impossible event and A∪B is an inevitable event, then event A and event B are mutually opposite events;

(4) When events A and B are mutually exclusive, the addition formula is satisfied: p (a ∪ b) = p (a)+p (b); If events A and B are opposite events, then A∪B is an inevitable event, so P(A

∪B)= P(A)+ P(B)= 1, so there is p (a) = 1-p (b).

2, the basic nature of probability:

1) The probability of inevitable events is 1, and the probability of impossible events is 0, so 0 ≤ p (a) ≤1; 2) When events A and B are mutually exclusive, the addition formula is satisfied: p (a ∪ b) = p (a)+p (b);

3) If events A and B are opposite, then A∪B is inevitable, so P(A∪B)= P(A)+ P(B)= 1, so there is P (A) =1-P (B);

4) The difference and connection between mutually exclusive events and opposing events, mutually exclusive events means that in an experiment, event A and event B will not happen at the same time, including three different situations: (1) Event A happens and event B doesn't happen; (2) Event A does not occur, but Event B does; (3) Event A and Event B are not simultaneous, but opposite.

Event means that event A and event B only occur once, including two situations; (1) Event A occurs, but event B does not; (2) Event B happens and Event A doesn't, which is a special case of mutually exclusive events.

Classical model of probability

(1) Conditions for using classical probability: finiteness of test results and equal possibility of all results. (2) the solution steps of classical probability; ① Find the total number of basic events;

② Find the number of' basic events' contained in event A, and then use the formula P(A)= 1

The number of basic events contained in

Total number of basic events

(3) the idea of transformation: the common' classical probability model': tossing coins, rolling dice, touching the ball (learning to count), drawing products, etc. Many probability models can be transformed.

Form the model above.

(4) If there is no sampling, it can be done out of order.

If there is a draw back, it should be sequential, and you can refer to the mode of two dice rolls.

Geometric model of probability

1, basic concept:

(1) Features of the geometric probability model: 1) There are infinitely many possible results (basic events) in the experiment; 2) The probability of each basic event is equal. (2) Probability formula of geometric probability:

The length (area or volume) of the region that constitutes event A.

P(A)= the length (area or volume) of the area formed by all test results;

(3) solving the geometric probability;

1. Determine what ratio it is: if the variable is selected as the length ratio within an interval or on a line segment, if the variable is selected as the area ratio in a plane figure, and if the variable is selected as a number.

What is the volume ratio in the body?

2. Find the key position and solve it.

(4) Special question type: Occasional question: If there are two variables in the question, use the method of combining numbers and shapes in rectangular coordinate system to solve it.

Symmetry knowledge points of mathematical circle

1, the axis symmetry of the circle

A circle is an axisymmetric figure, and every straight line passing through the center of the circle is its axis of symmetry.

2. The center of the circle is symmetrical

A circle is a central symmetrical figure with the center of the circle as the symmetrical center.

Knowledge points of mathematical inequality

1.( 1) Solving inequality is to find the solution set of inequality, and finally it must be expressed in the form of set; The endpoint value of inequality solution set is often the root of inequality corresponding equation or the endpoint value of inequality meaningful interval.

(2) What is the general idea of solving fractional inequalities? (moving the term into general division, the numerator and denominator decompose the factor, the coefficient of X becomes positive, and the standard root and odd number bounce back through even numbers);

(3) How to remove the absolute value from an inequality with two absolute values? (Generally, it is classified discussion, square transformation or substitution transformation according to the definition);

(4) The solutions of parametric inequalities are often classified as equivalent transformations, which need to be discussed in different categories if necessary. Note: discuss according to parameters, and finally explain their solution sets according to their values, but if discuss according to unknowns, finally find the union set.

2. When using important inequalities and variants to find the maximum value of a function, we must pay attention to A and B (or both A and B are non-negative). The condition of "equal sign" is that the product ab of a+b or one of them should be a fixed value (one is positive, two is fixed, three is equal and four is simultaneous).

3. The commonly used inequalities are: (selected according to the operation structure around the target inequality)

A, b, c, r, (if and only if, take the equal sign)

4. The methods of comparing sizes and proving inequalities mainly include difference comparison method, quotient comparison method, function property method, synthesis method and analysis method.

5. The nature of absolute inequality:

6. Inequality is established, can be established, just established, and so on.

The establishment of (1) constant

If the inequality is constant in the interval, it is equivalent to the interval.

If the inequality is constant in the interval, it is equivalent to the interval.

(2) the problem of establishment

(3) It is only a matter of establishment.

If the inequality is just established in the interval, it is equivalent to the solution set of the inequality.

If the inequality is just established in the interval, it is equivalent to the solution set of the inequality is,