1. 1 Write the sample space of the following random test and the set of sample points representing the following events.

(1) 10 products, 1 are unqualified products, and two of them are selected to get 1 unqualified products.

(2) There are two white balls, three black balls and four red balls in a pocket. Choose a ball from it, (1) get a white ball, (2) get a red ball.

Solution (1) is a qualified product for 9 times and a unqualified product for the second time, then

(2) Remember that two white balls are, three black balls are, and four red balls are,. Then {,,,,,}

(ⅰ) { , } (ⅱ) { , , , }

1.2 Select a student among the students in the Department of Mathematics, so that event A means that the selected student is a boy, event B means that the selected student is a junior three student, and event C means that the student is an athlete.

(1) The meaning of the narrative.

(2) Under what conditions?

(3) When is the relationship correct?

(4) When was it established?

The solution (1) event indicates that it should be a junior high school boy, not an athlete.

(2) Equivalence means that all athletes in the third grade have boys.

(3) When all athletes are third-grade students.

(4) When all the girls are in Grade Three and all the students in Grade Three are girls.

1.3 A worker produced a part, and the incident showed that the first part he produced was a qualified product (). Used to indicate the following events:

(1) All parts are unqualified;

(2) at least some of them are unqualified;

(3) Only some of them are unqualified;

(4) At least two parts are nonconforming products.

Solution (1); (2) ; (3) ;

(4) The original event "at least two parts are qualified products" can be expressed as:

1.4 proves the following types:

( 1) ;

(2)

(3) ;

(4)

(5)

(6)

Proof (1)-(4) Obviously, the proofs of (5) and (6) are similar to those of (10- 12) and (1.6) respectively.

1.5 Take any two cards with 2, 4, 6, 7, 8, 1 1, 12, 13 written on them, and combine the two numbers on the cards into a score to find the probability that the obtained score is the agreed score.

The total number of sample points is. The score obtained must be the denominator of the reduced score or two of 7, 1 1, 13, or one of 2, 4, 6, 8, 12 and 7,1,/kloc-0. therefore

.

1.6 has five line segments with lengths of 1, 3, 5, 7 and 9 respectively. Take any three of these five line segments and find the probability that these three line segments can form a triangle.

The total number of sample points is. The selected three line segments can form a triangle, and these three line segments must be greater than 3, 5, 7 or 3, 7, 9 or 5, 7, 9. So the event "three line segments can form a triangle" contains three sample points, so.

1.7 A child made a spelling game with 13 letters. If the arrangement of letters is random (equally possible), what is the probability of asking "the word' mathematician' is formed"?

Obviously, the total number of sample points is that "event" just constitutes "mathematician" including sample points. therefore

1.8 Put a red "car" and a black "car" on China's chessboard at random, and ask for the probability that they can just eat each other.

If the position of the red car is arbitrarily fixed, the black car can be in different positions, and when it and the red car are in the same row or one of the positions in the same row, they just "eat" each other. So the possibility is

1.9 The elevator on the floor of a building 10 got seven passengers on the ground floor. The elevator stops at every floor and passengers leave the elevator from the second floor. Assuming that the probability of each passenger leaving the elevator on which floor is equal, the probability that no two or more passengers leave on the same floor is found.

Each passenger can leave the elevator at any of the nine floors except the ground floor. There are currently 7 passengers, so the total number of sample points is. The event "no two or more passengers leave on the same floor" is equivalent to "from the ninth floor by the seventh floor, each passenger gets out of the elevator". So it contains a sample point.

1. 10 There are 10000 bicycles in a city, and their license plate numbers range from 0000 1 to 10000. What is the probability of "I accidentally bumped into a bicycle with the number 8 in the license plate number"?

There is a number 8 in the license plate number, which is obvious, so

-

Take any positive number from 1. 1 1 and find the probability of the following events:

(1) The last digit of the square of this number is1;

(2) The last digit of the fourth power of this number is1;

(3) The last two digits of the cube of this number are1;

Solution (1) The answer is.

(2) When the last digit of a number is one of 1, 3, 7 and 9, the last digit of the fourth power is 1, so the answer is

(3) The last two digits of the cube of a positive integer are determined by the last two digits of the number, so the sample space contains three sample points. If the event indicates that "the last two digits of the cube of this number are 1", then the last digit of this number must be 1. If the last two digits are, the last two digits of the cube of this number are the single digits of 1 and 3. The single digit of 3 needs to be 1, so only 765438 sample points are included.

1. 12 A man has six grasses in his hand, showing only his head and tail. Then ask another person to connect six heads in pairs and six tails in pairs. Find the probability that six grasses just connect in a circle after letting go. The above results are extended to the case of root grass.

Solve the problem of (1)6 grass. Take a head, it can be connected with one of the other five heads, then take another head, it can be connected with one of the other three unconnected heads, and finally connect the remaining two heads, so there are two kinds of connections between the head and the tail, so the total number of sample points is. For the head, there is still a connection method, but for the tail, if you take any tail, it can only be connected with the tails of the other four grasses that are not connected with its head. Take another tail, for example, it can only be connected with the tails of two other grasses that are not connected with its head, and finally connect the other two tails into a ring, so the connection method of the tails is as follows. So the number of sample points is, so

(2) The situation of root grass is similar to (1)

1. 13 randomly put exactly the same balls into a box (that is, after the balls are put into the box, only the number of balls in the box can be distinguished, but which ball has entered a box cannot be distinguished, which is also called indistinguishable). If the possibility of each method is equal, the probability of proving that there is a ball in the box specified by (1) is,

(2) The probability that there is exactly one box is,

(3) The probability that there happens to be a ball in the specified box is,

Solution.

1. 14 A bus arrives at a bus stop every 5 minutes, and the time for passengers to arrive at the bus stop is arbitrary. Find the probability that the waiting time of a passenger does not exceed 3 minutes.

The probability of the solution is

1. 15 If you choose any point, the probability that the proven area ratio is greater than IS.

If and only if the point falls within the cross-sectional area ratio, the cross-sectional area ratio is larger, so the probability is.

1. 16 The two ships will dock at the same berth and may arrive at any time day and night. The berthing time of two ships is 1 hour and two hours respectively, and the probability that a ship must wait for a period of time when berthing is obtained.

This solution is used to indicate the time when the first and second ships arrive at the berth. When a ship arrives at the berth, it must wait if and only if. So the possibility is

1. 17 Take any three points on the line segment and find:

(1) is between.

(2) The probability of forming a triangle.

Solution (1) (2)

1. 18 Draw equidistant parallel lines on the plane, throw a triangle into the plane at will, and find the probability that the triangle intersects the parallel lines.

These solutions are used to indicate that a vertex of a triangle coincides with a parallel line, an edge coincides with a parallel line, and two edges intersect with a parallel line. Obviously, the probability is. When two sides intersect parallel lines, it is obvious that … therefore.

[ ]

(Result of use case 1. 12)

1. 19 The probability of knowing the impossible event is zero. Now, is an event with zero probability necessarily an impossible event? Try to give examples.

An event with zero solution probability is not necessarily an impossible event. For example, randomly throw points onto a line segment with the length of 1. Then the probability of the event "the midpoint of this point hits" is equal to zero, but it is not an impossible event.

1.20 A and B take turns to touch a ball from the pocket containing a white ball and a black ball. A takes it first, and B takes it later. After each removal, the white ball was not put back until one of them took it away. Try to describe the probability space of this random phenomenon and find the probability that A or B gets the white ball first.

Solutions mean white, black and white, black and white, ...

Then the sample space {,,...}, and,

, ,…,

The probability of winning is+++…

The probability of B winning is++…

1.2 1 Let the probability sum of events be, and, and,,

Get a solution

,

1.22 Assuming two random events, it is proved that:

( 1) ;

(2) .

Proof (1)= 1

(2) The first inequality is obtained by the sum of (1), and the second and third inequalities are obtained by monotonicity and semi-additivity of probability.

1.23 For any random event,,, proves that:

certificate

1.24 There are three kinds of newspapers published in a city. Among the residents in this city, 45% subscribe to A newspaper, 35% subscribe to B newspaper, 30% subscribe to C newspaper, 10% subscribe to A newspaper and B newspaper at the same time, and 8% subscribe to A newspaper and C newspaper at the same time. Find the following percentage:

(1) only subscribed to newspaper a;

(2) Order only A and B newspapers;

(3) subscribing to only one newspaper;

(4) Subscribe to only two newspapers;

(5) Subscribe to at least one newspaper;

(6) Do not subscribe to any newspaper.

Solving an event is subscribing to a newspaper, an event is subscribing to a newspaper, and an event is subscribing to a newspaper.

( 1) = =30%

(2)

(3)

+ + = + + =73%

(4)

(5)

(6)

1.26 A student in a class took an oral test, and there were n papers. The test paper drawn by each student is put back after use. What is the probability that at least one test is not drawn after the test?

I didn't get the first paper. Requirements.

, ,……,

,……

therefore

1.27 Choose any item from the general expansion of determinant, and ask what is the probability that this item contains the main diagonal element?

In the determinant expansion of the order of the solution, any item with omitted symbols can be expressed as if and only if there is a cause in the arrangement of, and the item contains the principal diagonal element. Used to indicate that the event is "scheduled", that is, the first main diagonal element appears in an expanded project. rule

,……

therefore

1.29 It is known that a family has three children, one of whom is a girl. Find the probability of at least one boy (it is equally possible to assume that a child is a boy or a girl).

Interpretation refers to boys and girls respectively. The sample space is:

Among them, the sample points are arranged by age and gender. There is a girl and a boy.

1.30 One of the components is unqualified, so take two.

(1) If one of the selected products is unqualified, find the probability that the other product is also unqualified.

(2) If one of the selected products is qualified, find the probability that the other product is also unqualified.

If the solution (1) means "at least one of the selected products is unqualified" and "all the selected products are unqualified", then

(2) If "there is at least one qualified product in the selected product", it means "there is one qualified product and one unqualified product in the selected product". rule

1.3 1 people decide who gets a movie ticket by drawing lots. They drew lots in turn and asked:

(1) Knowing that no one has touched it before, find the probability that the first person has touched it;

(2) The probability that the first person comes into contact with it.

Solution means "the first person touched it".

( 1)

(2)

1.32 It is known that the probability of a hen laying an egg is 0, and the probability that each egg can hatch a chick is 0, which proves that the probability that a hen has just one offspring (that is, a chick) is 0.

The hen laid an egg, which means that the hen has just had the next generation.

1.33 A shooting group has 20 shooters, including 4 first-class shooters, 8 second-class shooters, 7 third-class shooters and 4 fourth-class shooters 1 person. The probability that the first, second, third and fourth-level shooters can enter the finals through selection is 0.9, 0.7, 0.5 and 0.2 respectively. Find the probability that any shooter in a group can enter the finals through selection.

It means "choose any shooter as a class", which means "choose any shooter to enter the final".

1.34 There are three machines A, B and C in a factory, the output accounts for 25%, 35% and 40% respectively, and the unqualified products account for 5%, 4% and 2% respectively. Now, any product is unqualified. What is the probability that this nonconforming product will be produced by Machine A, Machine B and Machine C respectively?

Any product is made by machine.

Any product is produced by machine B.

Any product is produced by machine C.

It means "taking any product is just unqualified"

Then through Bayesian formula:

1.35 The ratio of the number of lathes, drilling machines, grinders and planers in a factory is 9:3:2: 1, and the ratio of the probability that they need to be repaired in a certain period of time is 1:2:3: 1. When the machine tool needs maintenance, what is the probability of asking if the machine tool is a lathe?

Solutions,,,

, , ,

Derived from Bayesian formula

1.36 A friend came to visit from afar, and the probabilities of his coming by train, ship, car and plane were 0.3, 0.2, 0. 1 and 0.4 respectively. If he comes by train, ship or bus, the probability of being late is, and, but he won't be late by plane. As a result, he was late. What are the chances that he will come by train?

Friends come by train, friends come by boat, friends come by car, friends come by plane, and friends are late.

rule

1.37 proves that if three events,, are independent, then,, and are independent.

Proof (1)

=

(2)

(3) =

1.38 tries to explain that it must be established because it cannot be deduced.

Not set,,,

,,,, So,

but

Let 1.39 be an independent event, and find the probability of the following events:

(1) event did not occur;

(2) at least one event occurs;

(3) Just one thing happened.

Solution (1)

(2)

(3) .

1.40 The known events are independent and incompatible with each other, so we ask for it (note: a few).

On the one hand, on the other hand, at least one solution is equal to 0, so

1.4 1 The probability of a person's blood type is 0.46, 0.40, 0. 1 1, 0.03 respectively. Now, choose five people at will and find the probability of the following events.

(1) Two people are anthropomorphic, and the other three are other three blood types;

(2) Three people are humanoid and two people are humanoid;

(3) No one is artificial.

Solution (1) Choose two from five people, and there are three possibilities. There are three possibilities to choose a type from the remaining three people. Choose a type from the remaining two people, there are two possibilities, and the other is a type. The probability thus obtained is:

(2)

(3)

1.42 is equipped with two anti-aircraft guns, and the probability of each gun hitting the target is 0.6. What are the chances of firing a shell at the same time and hitting the plane? If an enemy plane invades the airspace, how many anti-aircraft guns are needed to hit it with a probability of more than 99%?

The first anti-aircraft gun fired a shell and hit the plane. Then,.

( 1)

(2) ,

Take it. It takes at least six anti-aircraft guns to fire 1 shells at the same time, so as to ensure a 99% probability of hitting the plane.

1.43 Do a series of independent experiments, and the probability of success in each experiment is to find the probability of failure before success.

It means "the number of failures before success", "the number of failures in previous experiments" and "the success of the first experiment"

rule

1.45 A mathematician has two boxes of matches, and each box has a match. Every time he uses a match, he chooses one from two boxes and draws one from them. Ask him the probability that he has used up one box and there are matches () in another box.

The definition is "there are matches left in box A", which means "there are matches left in box B", which means "I took them in box A for the first time" and "I took them in box B for the first time", which means that I took them from box A for the second time, that is, I took them in box A before, and the rest were in box B, so

From the symmetry point of view, the probability is: