A bag contains 5 white balls with the numbers $1,2,3,4,5$ written on them one each, and 5 black balls with the numbers $2,3,4,5,6$ written on them one each. When 2 balls are drawn simultaneously at random from the bag, what is the probability that the 2 balls drawn are either the same color or have the same number written on them? [3 points]
(1) $\frac { 7 } { 15 }$
(2) $\frac { 8 } { 15 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 11 } { 15 }$

In a deck of cards, what is the probability that when drawing 10 cards, none of them are hearts?

13. Among 9 randomly selected students, the probability that at least 2 students are born in the same month is $\_\_\_\_$ (assuming each month has the same number of days; round the result to 0.001)

A supermarket randomly selected 1000 customers and recorded their purchasing of four products: A, B, C, and D. The data is organized in the table below, where ``✓'' indicates purchase and ``×'' indicates no purchase.\n\n\n

\n\n\backslashbox{Number of Customers}{Product}	A	B	C	D
\n\n100	✓	×	✓	✓
\n\n217	×	✓	×	✓
\n\n200	✓	✓	✓	×
\n\n300	✓	×	✓	×
\n\n85	✓	×	×	×
\n\n98	×	✓	×	×
\n\n

\n\n\n(I) Estimate the probability that a customer purchases both B and C\n(II) Estimate the probability that a customer purchases exactly 3 of the four products A, B, C, and D\n(III) If a customer has purchased product A, which of products B, C, and D is the customer most likely to have purchased?

16. (This question is worth 12 points)
A shopping mall is holding a promotional lottery activity. After customers purchase goods of a certain amount, they can participate in the lottery. The lottery method is as follows: randomly draw 1 ball each from box A containing 2 red balls $\mathrm { A } _ { 1 } , \mathrm { A } _ { 2 }$ and 1 white ball B, and from box B containing 2 red balls $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 }$ and 2 white balls $\mathrm { b } _ { 1 } , \mathrm { b } _ { 2 }$. If both balls drawn are red, the customer wins; otherwise, the customer does not win. (I) List all possible outcomes of drawing balls using the ball labels. (II) Someone claims: Since there are more red balls than white balls in both boxes, the probability of winning is greater than the probability of not winning. Do you agree? Please explain your reasoning.

5. A bag contains 4 balls of identical shape and size, including 1 white ball, 1 red ball, and 2 yellow balls. If 2 balls are randomly drawn at once, then the probability that the 2 balls have different colors is $\_\_\_\_$ .

Chinese mathematician Chen Jingrun achieved world-leading results in research on Goldbach's conjecture. Goldbach's conjecture states that ``every even number greater than 2 can be expressed as the sum of two prime numbers'', such as $30 = 7 + 23$. Among prime numbers not exceeding 30, if two different numbers are randomly selected, the probability that their sum equals 30 is
A. $\frac { 1 } { 12 }$
B. $\frac { 1 } { 14 }$
C. $\frac { 1 } { 15 }$
D. $\frac { 1 } { 18 }$

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.

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.

Show by calculation that the probability of winning the game is $\frac { 1 } { 4 }$.

Let $G = ( S , A ) \in \Omega _ { n }$. Determine the probability $\mathbf { P } ( \{ G \} )$ of the elementary event $\{ G \}$ in terms of $p _ { n } , q _ { n } , N$ and $a = \operatorname { card } ( A )$. Then recover the fact that $\mathbf { P } \left( \Omega _ { n } \right) = 1$.

We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. We define the events:

• $A _ { n }$: "We obtain $p = q$".
• $C _ { n }$: "We obtain $p > q$".

Calculate $\mathbf { P } \left( A _ { n } \right)$ then $\mathbf { P } \left( C _ { n } \right)$.

147- We roll two dice together. With which probability is the sum of the two numbers rolled an odd number?
$$\frac{5}{12} \ (1) \hspace{2cm} \frac{4}{9} \ (2) \hspace{2cm} \frac{5}{9} \ (3) \hspace{2cm} \frac{7}{12} \ (4)$$

155- In a container there are 5 white marbles and 3 black marbles; in another container there are 4 white marbles and 2 black marbles. We randomly draw 4 marbles from each container. With which probability are the 4 marbles drawn all the same color?
$$0.12 \ (1) \hspace{2cm} 0.15 \ (2) \hspace{2cm} 0.18 \ (3) \hspace{2cm} 0.24 \ (4)$$
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147- Each of the numbers $1, 2, 3, 4, 5, 6$ is written on six equally likely balls. Consecutively, one ball is drawn from the box. What is the probability that an odd or even number appears among them?

(1) $0/1$

(2) $0/12$

(3) $0/15$

(4) $0/2$

147- We toss two coins and one die together. What is the probability that both heads ("ro") appear on the coins and 6 appears on the die?
(1) $\dfrac{3}{8}$ (2) $\dfrac{5}{8}$ (3) $\dfrac{5}{12}$ (4) $\dfrac{7}{12}$

147- Urn A contains 5 beads with odd digit numbers and Urn B contains 4 beads with non-zero even digit numbers. One bead is drawn from each urn. With which probability is the product of the two numbers greater than 10?
(1) $0.6$ (2) $0.65$ (3) $0.7$ (4) $0.75$

154- A fair coin is tossed repeatedly. What is the probability that the number 4 appears before the number 6?
(1) $\dfrac{1}{3}$ (2) $\dfrac{1}{2}$ (3) $\dfrac{2}{3}$ (4) $\dfrac{3}{4}$

A biased die, with faces numbered from 1 to 6, has the property that each even face appears with probability twice that of each odd face. Calculate the probabilities of obtaining, by rolling the die once, respectively:
－a prime number
－a number at least 3
－a number at most 3

26. There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is:
(A) $1 / 3$
(B) $1 / 6$
(C) $1 / 2$
(D) $1 / 4$

10. Two numbers are selected randomly from the set $S = \{ 1,2,3,4,5,6 \}$ without replacement one by one. The probability that minimum of the two numbers is less than 4 is :
(a) $[ 1 / 15 ]$
(b) $[ 14 / 15 ]$
(c) $[ 1 / 5 ]$
(d) $[ 4 / 5 ]$

12. A fair die is rolled. The probability that the first time 1 occurs at the even throw is:
(a) $1 / 6$ ... Powered By IITians
(b) $5 / 11$
(c) $6 / 11$
(d) $5 / 36$

A box $B _ { 1 }$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B _ { 2 }$ contains 2 white balls, 3 red balls and 4 black balls. A third box $B _ { 3 }$ contains 3 white balls, 4 red balls and 5 black balls.
If 1 ball is drawn from each of the boxes $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$, the probability that all 3 drawn balls are of the same colour is
(A) $\frac { 82 } { 648 }$
(B) $\frac { 90 } { 648 }$
(C) $\frac { 558 } { 648 }$
(D) $\frac { 566 } { 648 }$

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is
(A) $\frac{1}{2}$
(B) $\frac{1}{3}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$

Let $S = \left\{x \in (-\pi, \pi) : x \neq 0, \pm\frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3}\sec x + \operatorname{cosec} x + 2(\tan x - \cot x) = 0$ in the set $S$ is equal to
(A) $-\frac{7\pi}{9}$
(B) $-\frac{2\pi}{9}$
(C) $0$
(D) $\frac{5\pi}{9}$