1. Let sets $M = \{ x \mid 0 < x < 4 \}, N = \left\{ x \left\lvert \, \frac { 1 } { 3 } \leq x \leq 5 \right. \right\}$, then $M \cap N =$ A. $\left\{ x \left\lvert \, 0 < x \leq \frac { 1 } { 3 } \right. \right\}$ B. $\left\{ x \left\lvert \, \frac { 1 } { 3 } \leq x < 4 \right. \right\}$ C. $\{ x \mid 4 \leq x < 5 \}$ D. $\{ x \mid 0 < x \leq 5 \}$
2. Let $U = \{ 1,2,3,4,5,6 \} , A = \{ 1,3,6 \} , B = \{ 2,3,4 \}$. Then $A \cap \left( \complement_U B \right) =$ A. $\{ 3 \}$ B. $\{ 1,6 \}$ C. $\{ 5,6 \}$ D. $\{ 1,3 \}$ 【Answer】B 【Solution】 【Analysis】Use the definitions of intersection and complement to find $A \cap \left( \complement_U B \right)$. 【Detailed Solution】From the given conditions, $\complement_U B = \{ 1,5,6 \}$, so $A \cap \left( \complement_U B \right) = \{ 1,6 \}$, Therefore, the answer is: B.
Solution: Let the probabilities of events A, B, C, D be $P ( a ) , P ( b ) , P ( c ) , P ( d )$ respectively. Then $P ( a ) = \frac { 1 } { 6 } , P ( b ) = \frac { 5 } { 36 } , P ( d ) = \frac { 1 } { 36 }$. The probability that A and C occur simultaneously is $P ( a c ) = 0$; the probability that A and D occur simultaneously is $P ( a d ) = 0$; the probability that B and C occur simultaneously is $P ( b c ) = \frac { 1 } { 36 }$; the probability that C and D occur simultaneously is $P ( c d ) = 0$. The condition $P ( x y ) = P ( x ) P ( y )$ is satisfied by option B.
Let $U = \{ 1,2,3,4,5,6 \} , A = \{ 1,3,6 \} , B = \{ 2,3,4 \}$. Then $A \cap \left( \complement_U B \right) =$ A. $\{ 3 \}$ B. $\{ 1,6 \}$ C. $\{ 5,6 \}$ D. $\{ 1,3 \}$
5. From 7 integers from 2 to 8, two different numbers are randomly selected. The probability that these two numbers are coprime is A. $\frac { 1 } { 6 }$ B. $\frac { 1 } { 3 }$ C. $\frac { 1 } { 2 }$ D. $\frac { 2 } { 3 }$
A chess player plays one game each against three chess players A, B, and C, with the results of each game being independent. The probabilities that the player wins against A, B, and C are $p_1, p_2, p_3$ respectively, where $p_3 > p_2 > p_1 > 0$. Let $p$ denote the probability that the player wins two consecutive games. Then A. $p$ is independent of the order of games against A, B, and C B. $p$ is maximum when the player plays against A in the second game C. $p$ is maximum when the player plays against B in the second game D. $p$ is maximum when the player plays against C in the second game
Let $A = \{ x \mid x = 3k + 1 , k \in Z \} , B = \{ x \mid x = 3k + 2 , k \in Z \} , U$ be the set of integers, then $C_{U}(A \bigcap B) =$ A. $\{ x \mid x = 3k , k \in Z \}$ B. $\{ x \mid x = 3k - 1 , k \in Z \}$ C. $\{ x \mid x = 3k - 1 , k \in \mathrm{Z} \}$ D. $\varnothing$
Given proposition $p : \forall x \in \mathbf { R } , | x + 1 | > 1$; proposition $q : \exists x > 0 , x ^ { 3 } = x$, then A. Both $p$ and $q$ are true propositions B. Both $\neg p$ and $q$ are true propositions C. Both $p$ and $\neg q$ are true propositions D. Both $\neg p$ and $\neg q$ are true propositions
Person A and Person B each have four cards, with each card labeled with a number. Person A's cards are labeled with the numbers 1, 3, 5, 7, and Person B's cards are labeled with the numbers 2, 4, 6, 8. They play four rounds of competition. In each round, both players randomly select one card from their own cards and compare the numbers. The player with the larger number scores 1 point, and the player with the smaller number scores 0 points. Then each player discards the card used in that round (discarded cards cannot be used in subsequent rounds). The probability that Person A's total score after four rounds is at least 2 is $\_\_\_\_$ .
Let the universal set $U = \{x \mid x \text{ is a positive integer less than } 9\}$, and set $A = \{1,3,5\}$. Then the number of elements in $\complement_U A$ is A. $2$ B. $3$ C. $5$ D. $8$
In Sunnytown there are 6000 single-family houses, of which 2400 are equipped with a wood pellet heating system. Two thirds of the single-family houses with wood pellet heating have this combined with a solar thermal system. 50\% of all single-family houses are equipped with neither a wood pellet heating system nor a solar thermal system. (1a) [3 marks] Create a completely filled four-field table for the described situation. (1b) [2 marks] A randomly selected single-family house is equipped with a solar thermal system. What is the probability that it has a wood pellet heating system? The tree diagram shown represents a two-stage random experiment with events $A$ and $B$ as well as their complementary events $\bar { A }$ and $\bar { B }$. [Figure] (2a) [2 marks] Determine the value of $p$ so that event $B$ occurs in this random experiment with probability 0.3. (2b) [3 marks] Determine the maximum possible value that the probability of $B$ can assume. On a section of a lightly travelled country road, a maximum speed of 80 km/h is permitted. At one location on this section, the speed of passing cars is measured. In the following, only those journeys are considered where the drivers were able to choose their speed independently of one another. For the first 200 recorded journeys, the following distribution was obtained after classification into speed classes: [Figure] In 62\% of the 200 journeys, the driver was travelling alone, 65 of these solo drivers exceeded the speed limit. One journey is randomly selected from the 200 journeys. The following events are considered: $A$ : ``The driver was travelling alone.'' $S$ : ``The car was speeding.''
A wheel of fortune consists of five equally sized sectors. One of the sectors is labeled "0", one is labeled "1" and one is labeled "2"; the other two sectors are labeled "9". (1a) [2 marks] The wheel of fortune is spun four times. Calculate the probability that the numbers 2, 0, 1 and 9 are obtained in the specified order. (1b) [3 marks] The wheel of fortune is spun twice. Determine the probability that the sum of the numbers obtained is at least 11. (2) [3 marks] The random variable $X$ can only take the values 1, 4, 9 and 16. It is known that $P ( X = 9 ) = 0.2$ and $P ( X = 16 ) = 0.1$ as well as the expected value $E ( X ) = 5$. Using an approach for the expected value, determine the probabilities $P ( X = 1 )$ and $P ( X = 4 )$. (3) [2 marks] Given is a Bernoulli chain with length $n$ and success probability $p$. Explain that for all $k \in \{ 0 ; 1 ; 2 ; \ldots ; n \}$ the relationship $B ( n ; p ; k ) = B ( n ; 1 - p ; n - k )$ holds. A company organizes trips with an excursion ship that has space for 60 passengers.
In a municipality there are 6250 households, of which 2250 have a fast internet connection. Two thirds of the households that have a fast internet connection also have a subscription to a streaming service. $46 \%$ of all households have neither a fast internet connection nor a subscription to a streaming service. Consider the following events:\ $A$ : ``A randomly selected household has a fast internet connection.''\ $B$ : ``A randomly selected household has a subscription to a streaming service.''\ Create a completely filled four-field table for the described situation and check whether the events $A$ and $B$ are stochastically independent.
On a Saturday morning, four families arrive one after another at the entrance area of an amusement park. Each of the four families pays at one of six cashiers, whereby it is assumed that each cashier is chosen with equal probability. Describe in the context of the problem two events $A$ and $B$ whose probabilities can be calculated using the following terms: $P ( A ) = \frac { 6 \cdot 5 \cdot 4 \cdot 3 } { 6 ^ { 4 } } ; P ( B ) = \frac { 6 } { 6 ^ { 4 } }$
The amusement park holds a game of chance in which entrance tickets to the amusement park can be won. At the beginning of the game, one rolls a die whose faces are numbered with the numbers 1 to 6. If one obtains the number 6, one may then spin a wheel of fortune with three sectors once (see schematic diagram). If sector $K$ is obtained, one wins a child ticket worth 28 euros; for sector $E$, an adult ticket worth 36 euros. For sector $N$, one wins nothing. The central angle of sector $N$ is $160 ^ { \circ }$. The sizes of sectors $K$ and $E$ are chosen such that the average winnings per game amount to three euros. Determine the size of the central angles of sectors $K$ and $E$.
Determine, assuming that the proportion of employees with a job ticket is the same at both locations, the probability that a randomly selected employee of the automotive supplier works at location B and does not have a job ticket.