When a coin is tossed 5 times, what is the probability that the product of the number of heads and the number of tails is 6? [3 points]
(1) $\frac { 5 } { 8 }$
(2) $\frac { 9 } { 16 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 16 }$
(5) $\frac { 3 } { 8 }$

There are 4 white balls with the numbers $1, 2, 3, 4$ written on them and 3 black balls with the numbers $4, 5, 6$ written on them. When these 7 balls are randomly arranged in a line, the probability that the balls with the same number do not lie adjacent to each other is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

There are 6 cards with the numbers $1, 2, 3, 4, 5, 6$ written on them, one number per card. When all 6 cards are arranged in a row in random order using each card exactly once, find the probability that the sum of the two numbers on the cards at both ends is at most 10. [3 points]
(1) $\frac{8}{15}$
(2) $\frac{19}{30}$
(3) $\frac{11}{15}$
(4) $\frac{5}{6}$
(5) $\frac{14}{15}$

15. (13 points) Three table tennis associations have 27, 9, and 18 members respectively. Using stratified sampling, 6 athletes are selected from these three associations to participate in a competition. (I) Find the number of athletes to be selected from each of the three associations respectively; (II) The 6 selected athletes are numbered $A _ { 1 } , A _ { 2 } , A _ { 3 } , A _ { 4 } , A _ { 5 } , A _ { 6 }$ respectively. Two athletes are randomly selected from these 6 athletes to participate in a doubles match.
(i) List all possible outcomes using the given numbering;
(ii) Let event $A$ be ``at least one of the two athletes numbered $A _ { 5 }$ and $A _ { 6 }$ is selected''. Find the probability of event $A$ occurring.

10. Three 1's and two 0's are randomly arranged in a row. The probability that the two 0's are not adjacent is
A. 0.3
B. 0.5
C. 0.6
D. 0.8

10. Arrange 4 ones and 2 zeros randomly in a row. The probability that the 2 zeros are not adjacent is [Figure]
A. $\frac{1}{3}$
B. $\frac{2}{5}$
C. $\frac{2}{3}$
D. $\frac{4}{5}$

For every integer $n \geqslant 2$, let $p_n$ be the probability that a uniformly random permutation of $\llbracket 1, n \rrbracket$ is alternating up (with $p_0 = p_1 = 1$). Show that the sequence $(p_n)$ tends to 0. Give an equivalent of $p_{2n+1}$ as $n$ tends to infinity.

There are five students $S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }$ and $S _ { 5 }$ in a music class and for them there are five seats $R _ { 1 } , R _ { 2 } , R _ { 3 } , R _ { 4 }$ and $R _ { 5 }$ arranged in a row, where initially the seat $R _ { i }$ is allotted to the student $S _ { i } , i = 1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats. The probability that, on the examination day, the student $S _ { 1 }$ gets the previously allotted seat $R _ { 1 }$, and NONE of the remaining students gets the seat previously allotted to him/her is
(A) $\frac { 3 } { 40 }$
(B) $\frac { 1 } { 8 }$
(C) $\frac { 7 } { 40 }$
(D) $\frac { 1 } { 5 }$

There are five students $S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }$ and $S _ { 5 }$ in a music class and for them there are five seats $R _ { 1 } , R _ { 2 } , R _ { 3 } , R _ { 4 }$ and $R _ { 5 }$ arranged in a row, where initially the seat $R _ { i }$ is allotted to the student $S _ { i } , i = 1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats. For $i = 1,2,3,4$, let $T _ { i }$ denote the event that the students $S _ { i }$ and $S _ { i + 1 }$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T _ { 1 } \cap T _ { 2 } \cap T _ { 3 } \cap T _ { 4 }$ is
(A) $\frac { 1 } { 15 }$
(B) $\frac { 1 } { 10 }$
(C) $\frac { 7 } { 60 }$
(D) $\frac { 1 } { 5 }$

Let $X$ be the set of all five digit numbers formed using $1,2,2,2,4,4,0$. For example, 22240 is in $X$ while 02244 and 44422 are not in $X$. Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of $38p$ is equal to

Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is :
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 1 } { 3 }$

Let $P ( X )$ denote the probability of event $X$ occurring, and $P ( X \mid Y )$ denote the probability of event $X$ occurring given that event $Y$ has occurred. There are 7 balls of the same size: 2 black balls, 2 white balls, and 3 red balls arranged in a row. Let event $A$ be the event that the 2 black balls are adjacent, event $B$ be the event that the 2 black balls are not adjacent, and event $C$ be the event that no two red balls are adjacent. Select the correct options.
(1) $P ( A ) > P ( B )$
(2) $P ( C ) = \frac { 2 } { 7 }$
(3) $2 P ( C \mid A ) + 5 P ( C \mid B ) < 2$
(4) $P ( C \mid A ) > 0.2$
(5) $P ( C \mid B ) > 0.3$