Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. For a graph $H = \left( S _ { H } , A _ { H } \right)$ with $S _ { H } \subset \llbracket 1 , n \rrbracket$, the random variable $X_H$ is defined by: $$\forall G \in \Omega _ { n } \quad X _ { H } ( G ) = \begin{cases} 1 & \text { if } H \subset G \\ 0 & \text { otherwise } \end{cases}$$ Show that $$\mathbf { E } \left( X _ { H } \right) = p _ { n } ^ { a _ { H } } .$$

Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. Let $\mathcal{C}_0$ be the set of copies of $G_0$ whose vertices are included in $\llbracket 1, n \rrbracket$: $$\mathcal { C } _ { 0 } = \left\{ H \mid H \text { is a copy of } G _ { 0 } \text { and } H = \left( S _ { H } , A _ { H } \right) \text { with } S _ { H } \subset \llbracket 1 , n \rrbracket \right\}$$ Let $S _ { 0 } ^ { \prime }$ be a fixed set of cardinality $s _ { 0 }$. We denote by $c _ { 0 }$ the number of graphs whose vertex set is $S _ { 0 } ^ { \prime }$ and which are copies of $G _ { 0 }$. Express the cardinality of $\mathcal { C } _ { 0 }$ using $c _ { 0 }$ and using a simple upper bound for $c _ { 0 }$, justify that the cardinality of $\mathcal { C } _ { 0 }$ is less than $n ^ { s _ { 0 } }$.

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:

• $E _ { n }$: "We obtain $( p , q ) \in E _ { 1 } \cup E _ { 2 } \cup E _ { 3 }$".
• $A _ { n }$: "We obtain $p = q$".
• $B _ { n }$: "We obtain $q > p$ and $q$ is divisible by $p$".
• $C _ { n }$: "We obtain $p > q$".

where $E_1 = \{(p,q)\in(\mathbf{N}^*)^2: p=q\}$, $E_2 = \{(p,q)\in(\mathbf{N}^*)^2: pq\}$.
Justify that the set $\left\{ A _ { n } , B _ { n } , C _ { n } \right\}$ forms a partition of $E _ { n }$.

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

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 $p = q$".
• $B _ { n }$: "We obtain $q > p$ and $q$ is divisible by $p$".

Show that $$\mathbf { P } \left( B _ { n } \right) = \frac { 1 } { n ^ { 2 } } \sum _ { p = 1 } ^ { n } \left\lfloor \frac { n } { p } \right\rfloor - \frac { 1 } { n } ,$$ and deduce $\mathbf { P } \left( A _ { n } \cup B _ { n } \right)$.

We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. Using the result $H_n \sim \ln n$ as $n \to +\infty$, show that $$\mathbf { P } \left( A _ { n } \cup B _ { n } \right) \sim \frac { \ln n } { n } \quad ( n \rightarrow + \infty )$$

We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. The event $E_n$ is defined as "We obtain $(p,q) \in E_1 \cup E_2 \cup E_3$".
Using the result $\mathbf{P}(A_n \cup B_n) \sim \dfrac{\ln n}{n}$ as $n \to +\infty$, deduce $$\lim _ { n \rightarrow + \infty } \mathbf { P } \left( E _ { n } \right) .$$

148. If $A$ and $B$ are two events from sample space $S$ such that $P(A) = 0.6$, $P(B) = 0.7$, and $P(A \cap B') = 0.2$, then $P(A' \cap B)$ is equal to?
(1) $0.1$ (2) $0.3$ (3) $0.4$ (4) $0.5$

154. Five white marbles numbered 1 to 5 and five black marbles numbered 1 to 5 are placed in two separate containers. Two marbles are drawn at random from each container. If the sum of the two marbles from each container is 6, what is the probability that both marbles are the same color?
(1) $\dfrac{2}{5}$ (2) $\dfrac{4}{9}$ (3) $\dfrac{5}{9}$ (4) $\dfrac{3}{5}$

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) $0/2$

155. In two containers there are respectively 24 and 18 identical balls. In the first container there are 6 white balls and in the second container there are 3 white balls. From the first container 7 balls and from the second container 5 balls are randomly drawn and placed in another container. Then from the last container one ball is drawn. What is the probability that this ball is white?
(1) $\dfrac{13}{72}$ (2) $\dfrac{7}{36}$
(3) $\dfrac{15}{72}$ (4) $\dfrac{31}{144}$
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148. From the set of consecutive integers $\{300, 301, \ldots, 51\}$... wait: $\{300, 301, \ldots, 51\}$, i.e., $\{300, 301, \ldots, 451\}$, a number is chosen at random. What is the probability that this number is divisible by 6 but not by 7 and not by 42?
(1) $0/24$ (2) $0/26$ (3) $0/28$ (4) $0/31$

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) $0.75$

148- Three people are working on decoding a message. Their probabilities of success are $\frac{2}{3}$, $\frac{3}{4}$, and $\frac{1}{2}$ respectively. What is the probability that at least one of them succeeds?
(1) $\dfrac{19}{24}$ (2) $\dfrac{5}{6}$ (3) $\dfrac{11}{12}$ (4) $\dfrac{23}{24}$

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) $\dfrac{3}{4}$

145- A two-digit natural number is chosen at random. What is the probability that the chosen number is a multiple of 3 or 5?
\[ (1)\quad \frac{2}{5} \qquad (2)\quad \frac{3}{5} \qquad (3)\quad \frac{7}{15} \qquad (4)\quad \frac{8}{15} \]

146- We have three urns. The first urn contains 9 white and 4 black marbles, the second contains 9 black and 4 white marbles, and the third contains 5 white and 5 black marbles. One marble is randomly drawn from one urn. What is the probability that at least one of these two marbles is black?
\[ (1)\quad \frac{1}{3} \qquad (2)\quad \frac{11}{18} \qquad (3)\quad \frac{25}{36} \qquad (4)\quad \frac{13}{18} \]
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130- We write the numbers 1 to 21 each on a card and place them in a bag. We then randomly draw two cards one after another without replacement and place them aside. The set of all possible outcomes forms set $A$. We select one number from set $A$. What is the probability that the selected number is divisible by 6?
(1) $\dfrac{13}{84}$ (2) $\dfrac{65}{417}$ (3) $\dfrac{11}{70}$ (4) $\dfrac{67}{417}$

136. In the first jar there are 3 blue marbles and 6 red marbles, and in the second jar there are 4 blue marbles and 5 red marbles. Two coins are tossed. If the total number of heads is more than 9, one marble is taken from the first jar and added to the second jar. Otherwise, one marble is taken from the second jar and added to the first jar. Now one marble is selected from the jar with more marbles. What is the probability that this marble is red?
(1) $\dfrac{157}{270}$ (2) $\dfrac{165}{270}$ (3) $\dfrac{173}{270}$ (4) $\dfrac{185}{270}$

126-- In a random experiment, $S=\{x,y,z\}$ is a sample space. If $P(x)$, $P(y)$, and $P(z)$ form a geometric sequence and together they are less than one unit and their geometric mean is $\dfrac{1}{5}$, then the smallest simple event in $S$ is how much?
(1) $\dfrac{2-\sqrt{2}}{5}$ (2) $\dfrac{2-\sqrt{2}}{5}$ (3) $\dfrac{2-\sqrt{3}}{10}$ (4) $\dfrac{2-\sqrt{3}}{10}$

22 -- In a group of 150 students, 40 bought only a ticket for film ``A'' and 75 bought only a ticket for film ``B''. If $P(A)$ and $P(B)$ are respectively the probabilities of buying a ticket for films ``A'' and ``B'', what is the maximum value of $\dfrac{P(A)}{P(B)}$?
(1) $\dfrac{15}{29}$ (2) $\dfrac{38}{45}$ (3) $\dfrac{8}{15}$ (4) $\dfrac{15}{22}$

24 -- A device is designed so that it randomly receives one of two letters A or B as input and passes through three stages. At each stage, the input letter is printed with probability $\frac{1}{r}$ without change, or it moves to the next stage changed to the other letter. If the probability of selecting letter A as input is 2 times that of letter B, with what probability is ``A'' printed by the device equal to the probability of the input letter being A?
(1) $\dfrac{14}{23}$ (2) $\dfrac{21}{22}$ (3) $\dfrac{9}{41}$ (4) $\dfrac{17}{41}$
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