Establish the following identity: for $( s , i , r ) \in E$, for all $k \in \{ 0 , \cdots , s \}$,
$$\mathbf { P } \left( \Delta \tilde { S } _ { n } = - k \mid \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right) = \binom { s } { k } ( p ( i ) ) ^ { k } ( 1 - p ( i ) ) ^ { s - k }$$
where $p(i)$ is the probability for a susceptible person to be infected during the day (as found in question 21), and the $s$ susceptible persons act independently.

Let $n$ be a positive integer and $t \in ( 0,1 )$. Then $\sum _ { r = 0 } ^ { n } r \binom { n } { r } t ^ { r } ( 1 - t ) ^ { n - r }$ equals
(A) $n t$
(B) $( n - 1 ) ( 1 - t )$
(C) $n t + ( n - 1 ) ( 1 - t )$
(D) $\left( n ^ { 2 } - 2 n + 2 \right) t$.

There are 30 True or False questions in an examination. A student knows the answer to 20 questions and guesses the answers to the remaining 10 questions at random. What is the probability that the student gets exactly 24 answers correct?
(A) $\frac{105}{2^9}$
(B) $\frac{105}{2^8}$
(C) $\frac{105}{2^{10}}$
(D) $\frac{4}{2^{10}}$

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is

Let $C _ { 1 }$ and $C _ { 2 }$ be two biased coins such that the probabilities of getting head in a single toss are $\frac { 2 } { 3 }$ and $\frac { 1 } { 3 }$, respectively. Suppose $\alpha$ is the number of heads that appear when $C _ { 1 }$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C _ { 2 }$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x ^ { 2 } - \alpha x + \beta$ are real and equal, is
(A) $\frac { 40 } { 81 }$
(B) $\frac { 20 } { 81 }$
(C) $\frac { 1 } { 2 }$
(D) $\frac { 1 } { 4 }$

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is
(1) $1 / 729$
(2) $8 / 9$
(3) $8 / 729$
(4) $8 / 243$

A multiple choice examination has 5 questions. Each question has three alternative answers out of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is:
(1) $\frac{11}{3^5}$
(2) $\frac{10}{3^5}$
(3) $\frac{17}{3^5}$
(4) $\frac{13}{3^5}$

An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is
(1) $\frac { 496 } { 729 }$
(2) $\frac { 192 } { 729 }$
(3) $\frac { 240 } { 729 }$
(4) $\frac { 256 } { 729 }$

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:
(1) $\dfrac{12}{5}$
(2) $6$
(3) $4$
(4) $\dfrac{6}{25}$

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:
(1) $\frac { 6 } { 25 }$
(2) 6
(3) 4
(4) $\frac { 12 } { 5 }$

The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least $90 \%$ is:
(1) 2
(2) 4
(3) 5
(4) 3

In a bombing attack, there is $50\%$ chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that the probability of the target being destroyed is at least $0.99$, is ...

Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is: (1) $\frac { 5 } { 8 }$ (2) $\frac { 1 } { 8 }$ (3) $\frac { 5 } { 16 }$ (4) 1

If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is:
(1) $\frac { 33 } { 2 ^ { 32 } }$
(2) $\frac { 33 } { 2 ^ { 29 } }$
(3) $\frac { 33 } { 2 ^ { 28 } }$
(4) $\frac { 33 } { 2 ^ { 27 } }$

The mean and variance of a binomial distribution are $\alpha$ and $\frac { \alpha } { 3 }$ respectively. If $P ( X = 1 ) = \frac { 4 } { 243 }$, then $P ( X = 4$ or $5 )$ is equal to:
(1) $\frac { 5 } { 9 }$
(2) $\frac { 64 } { 81 }$
(3) $\frac { 16 } { 27 }$
(4) $\frac { 145 } { 243 }$

Let $X$ have a binomial distribution $B ( n , p )$ such that the sum and the product of the mean and variance of $X$ are 24 and 128 respectively. If $P ( X > n - 3 ) = \frac { k } { 2 ^ { n } }$, then $k$ is equal to
(1) 528
(2) 529
(3) 629
(4) 630

Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is
(1) $\frac { 46 } { 6 ^ { 4 } }$
(2) $\frac { 275 } { 6 ^ { 5 } }$
(3) $\frac { 41 } { 5 ^ { 5 } }$
(4) $\frac { 36 } { 5 ^ { 4 } }$

In a binomial distribution $B(n, p)$, the sum and product of the mean and variance are 5 and 6 respectively, then $6(n + p - q)$ is equal to:
(1) 51
(2) 52
(3) 53
(4) 50

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac { 1 } { 3 }$ and $\frac { 2 } { 3 }$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $\mathrm { P } ( | x - y | \leq 2 )$ is $p$, then $3 ^ { 9 } p$ equals $\_\_\_\_$

Let P be a point in a plane with a coordinate system that is initially located at the origin $(0,0)$ and moves in the plane according to the following rule:
One dice is thrown. When the number on the dice is a multiple of three, point P moves 1 unit in the positive direction of the $x$-axis, and when the number on the dice is not a multiple of three, point P moves 1 unit in the positive direction of the $y$-axis.
Assume that the dice is thrown four times.
(1) The probability that P reaches point $(3,1)$ is $\frac{\mathbf{A}}{\mathbf{BC}}$.
(2) Altogether, the number of the points which P can reach is $\mathbf{D}$, and the coordinates of these points can be expressed in terms of an integer $k$ as
$$(k,\, \mathbf{E} - k) \quad (\mathbf{F} \leq k \leq \mathbf{G}).$$
Let us denote the probability that P can reach a given point $(k, \mathbf{E} - k)$ by $p_k$. Then the maximum value of $p_k$ is $\frac{\mathbf{HI}}{\mathbf{HI}}$, and the minimum value of $p_k$ is $\frac{\mathbf{J}}{\mathbf{BC}}$.
(3) The probability that $P$ passes through point $(1,1)$ and reaches point $(2,2)$ is $\frac{\mathbf{KL}}{\mathbf{BC}}$.

Q1 In a box, there are $n$ red balls and $(20 - n)$ white balls, where $0 < n < 20$. In each trial, a ball is taken out of the box, its color is examined, and it is returned to the box.
(1) Let $x$ be the probability that the ball taken out in one trial is red. Then, $x = \frac{n}{\mathbf{AB}}$.
(2) Let $p$ be the probability that in two trials a white ball is taken out at least once. Then $p$ can be expressed as $p = \mathbf{C} - x^{\mathbf{D}}$, where $x$ is the $x$ of (1).
(3) Let $q$ be the probability that in four trials a white ball is taken out at least twice. Then $q$ can be expressed as
$$q = \mathbf{E} - \mathbf{H}x^{\mathbf{G}} + \mathbf{H}x^{\mathbf{I}},$$
where $x$ is the $x$ of (1).
(4) For $p$ and $q$ of (2) and (3), we are to find the maximum value of $n$ such that $p < q$.
From the inequality $p < q$, we obtain the inequality
$$\mathbf{J}x^2 - \mathbf{K} \square \text{ } \square$$
When we solve this, we have
$$x < \frac{1}{\mathbf{L}}$$
Thus the maximum value of $n$ is $\square\mathbf{M}$.

For a game, each of two people, A and B , has a bag containing three cards on which the numbers 1, 2, and 3 are written, each number on a different card. In the game, A and B each take out one card from their own bag and compare the numbers. If the numbers are the same, the game is a draw. If the numbers are different, the person with the greater number wins.
(1) For a single game the probability of a draw is $\frac { \mathbf { L } } { \mathbf { M } }$.
(2) If this game is successively played four times, replacing the cards after each game, let us find the probabilities for the following.
(i) The probability that A wins three times or more is $\frac { \mathbf { N } } { \mathbf{O} }$.
(ii) The probability that A wins once and loses once and two games are draws is $\frac { \mathrm { P } } { \mathrm { QR } }$.
(iii) The probability that the number of times that A wins and the number of times that B wins are the same is $\frac { \mathbf { S T } } { \mathbf { U V } }$. Hence, the probability that the number of times that A wins is greater than the number of times that B wins is $\frac { \mathbf { W } \mathbf { X } } { \mathbf{UV} }$.

A biased coin has probability $\frac { 1 } { 3 }$ of showing heads and probability $\frac { 2 } { 3 }$ of showing tails. On a coordinate plane, a game piece moves to the next position based on the result of flipping this coin, according to the following rules: (I) If heads appears, the piece moves from its current position in the direction and distance of vector $( - 1,2 )$ to the next position; (II) If tails appears, the piece moves from its current position in the direction and distance of vector $( 1,0 )$ to the next position. For example: If the game piece is currently at coordinates $( 2,4 )$ and tails appears, the piece moves to coordinates $( 3,4 )$. Suppose the game piece starts at the origin $( 0,0 )$ and, according to the above rules, flips the coin 6 times consecutively, with each flip being independent. After 6 moves, the game piece is most likely to stop at coordinates (12), (13)).

It is desired to place 4 identical chess rooks on a $5 \times 5$ chessboard. Since rooks can capture pieces in the same row or column, the placement rule is that at most one rook can be placed in each row and each column. Given that rooks are not placed in the first, third, and fifth squares of the first row (as shown by the crossed squares in the diagram), how many ways are there to place the rooks?
(1) 216
(2) 240
(3) 288
(4) 312
(5) 360

A store launches a lottery activity offering four different styles of fruit figurines as prizes. Each lottery draw yields 1 figurine, and each style has an equal probability of being drawn. A person decides to draw 4 times. What is the probability that he draws exactly 3 different styles of figurines?
(1) $\frac { 5 } { 16 }$
(2) $\frac { 3 } { 8 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 9 } { 16 }$
(5) $\frac { 5 } { 8 }$