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.

154- In a box there are $8$ light bulbs, two of which are defective. The bulbs are tested randomly one by one and the good bulb is set aside, until the first defective bulb is found. In the third test, what is the probability of finding the first defective bulb?
(1) $\dfrac{5}{28}$ (2) $\dfrac{4}{21}$ (3) $\dfrac{3}{14}$ (4) $\dfrac{5}{21}$

155- In a bag there are $5$ white beads, $4$ blue beads, and $3$ red beads. We draw three beads from the bag. What is the probability that at most $2$ of the drawn beads are the same color?
(1) $\dfrac{17}{22}$ (2) $\dfrac{19}{22}$ (3) $\dfrac{39}{44}$ (4) $\dfrac{41}{44}$
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155- Given $P(X = x) = \dfrac{\dbinom{5}{x}\dbinom{4}{r-x}}{a}$\,; $x = 0, 1, 2, 3$, for which value of $a$ is this a probability function?
(1) $48$ (2) $56$ (3) $64$ (4) $84$
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Football teams $T _ { 1 }$ and $T _ { 2 }$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T _ { 1 }$ winning, drawing and losing a game against $T _ { 2 }$ are $\frac { 1 } { 2 } , \frac { 1 } { 6 }$ and $\frac { 1 } { 3 }$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T _ { 1 }$ and $T _ { 2 }$, respectively, after two games.
$P ( X > Y )$ is
(A) $\frac { 1 } { 4 }$
(B) $\frac { 5 } { 12 }$
(C) $\frac { 1 } { 2 }$
(D) $\frac { 7 } { 12 }$

Football teams $T _ { 1 }$ and $T _ { 2 }$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T _ { 1 }$ winning, drawing and losing a game against $T _ { 2 }$ are $\frac { 1 } { 2 } , \frac { 1 } { 6 }$ and $\frac { 1 } { 3 }$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T _ { 1 }$ and $T _ { 2 }$, respectively, after two games.
$P ( X = Y )$ is
(A) $\frac { 11 } { 36 }$
(B) $\frac { 1 } { 3 }$
(C) $\frac { 13 } { 36 }$
(D) $\frac { 1 } { 2 }$

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 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) $\frac { 12 } { 5 }$

In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :
(1) $\frac { 400 } { 3 }$ gain
(2) $\frac { 400 } { 9 }$ gain
(3) $\frac { 400 } { 3 }$ loss
(4) 0

An unbiased coin is tossed 5 times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k = 3, 4, 5$, otherwise $X$ takes the value $-1$. Then the expected value of $X$, is
(1) $\frac { 3 } { 16 }$
(2) $\frac { 1 } { 8 }$
(3) $- \frac { 3 } { 16 }$
(4) $- \frac { 1 } { 8 }$

In a box, there are 20 cards, out of which 10 are labelled as $A$ and the remaining 10 are labelled as $B$. Cards are drawn at random, one after the other and with replacement, till a second $A$ card is obtained. The probability that the second $A$ card appears before the third $B$ card is:
(1) $\frac { 9 } { 16 }$
(2) $\frac { 11 } { 16 }$
(3) $\frac { 13 } { 16 }$
(4) $\frac { 15 } { 16 }$

A six faced die is biased such that $3 \times P ($ a prime number $) = 6 \times P ($ a composite number $) = 2 \times P ( 1 )$. Let $X$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of $X$ is
(1) $\frac { 3 } { 11 }$
(2) $\frac { 5 } { 11 }$
(3) $\frac { 7 } { 11 }$
(4) $\frac { 8 } { 11 }$

A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let $X$ be the number of white balls, among the drawn balls. If $\sigma ^ { 2 }$ is the variance of $X$, then $100 \sigma ^ { 2 }$ is equal to $\_\_\_\_$.

Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N - 2 , \sqrt { 3 N } , N + 2$ are in geometric progression be $\frac { k } { 48 }$. Then the value of $k$ is
(1) 2
(2) 4
(3) 16
(4) 8

A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is
(1) $\frac { 5 } { 6 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 5 } { 11 }$
(4) $\frac { 6 } { 11 }$

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar { X }$ and $\bar { Y }$ are the means of $X$ and $Y$ respectively, then $7 \bar { X } + 4 \bar { Y }$ is equal to $\_\_\_\_$

Let $\mathrm { A } = \left[ \mathrm { a } _ { i j } \right]$ be a $2 \times 2$ matrix such that $\mathrm { a } _ { i j } \in \{ 0,1 \}$ for all $i$ and $j$. Let the random variable X denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is :
(1) $\frac { 3 } { 4 }$
(2) $\frac { 5 } { 8 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 4 }$

Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denote the number of defective oranges, then the variance of $x$ is
(1) $28/75$
(2) $18/25$
(3) $26/75$
(4) $14/25$

A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma ^ { 2 }$ denote the mean and variance of $X$, then the value of $64 \left( \mu + \sigma ^ { 2 } \right)$ is:
(1) 51
(2) 64
(3) 32
(4) 48

Q90. Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar { X }$ and $\bar { Y }$ are the means of $X$ and $Y$ respectively, then $7 \bar { X } + 4 \bar { Y }$ is equal to $\_\_\_\_$

ANSWER KEYS

\begin{tabular}{|l|l|l|l|} \hline 1. (2) & 2. (2) & 3. (3) & 4. (1) \hline 9. (2) & 10. (2) & 11. (1) & 12. (1) \hline 17. (4) & 18. (2) & 19. (2) & 20. (1) \hline 25. (12) & 26. (748) & 27. (4) & 28. (3) \hline 33. (2) & 34. (1) & 35. (1) & 36. (4) \hline

There are two boxes, A and B.
In box A, there are three cards on which the number 0 is written, two cards on which the number 2 is written, and one card on which the number 3 is written.
In box B, there are two cards on which the number 1 is written, and three cards on which the number 2 is written.
Take two cards together from box A and one card from box B. Denote the product of the numbers on the three cards by $X$.
The total number of values which $X$ can take is A. The maximum value which $X$ can take is $\mathbf{BC}$. The minimum value which $X$ can take is D.
The probability that $X = \mathrm{BC}$ is $\frac{\mathbf{E}}{\mathrm{F}}$, and the probability that $X = \square\mathrm{D}$ is $\frac{\mathbf{H}}{\mathbf{I}}$.

3. A box contains three red balls and three white balls. A lottery game involves randomly drawing two balls simultaneously from the box. If the two balls are different colors, the player wins 100 dollars; if the two balls are the same color, there is no prize. What is the expected value of the prize for this game?
(1) 20 dollars
(2) 30 dollars
(3) 40 dollars
(4) 50 dollars
(5) 60 dollars

If the possible values of random variable $X$ are $1 , 2 , 3 , 4$ , and the probability $P ( X = k )$ is proportional to $\frac { 1 } { k }$ , then the probability $P ( X = 3 )$ is $\frac { \text{(9)(11)} } { \text{(10)(11)} }$ . (Reduce to lowest terms)

There is a game involving moving a game piece on a number line. The way to move the piece is determined by rolling a fair die, with the following rules: (I) When the die shows 1 point, the piece does not move. (II) When the die shows 3 or 5 points, the piece moves left (negative direction) by ``that point number minus 1'' units. (III) When the die shows an even number, the piece moves right (positive direction) by ``half of that point number'' units. On the first die roll, the piece starts at the origin. From the second roll onwards, the piece starts from the position it was in after the previous roll. For example, if two die rolls result in 5 points and 2 points respectively, the piece first moves left 4 units to coordinate $-4$, then moves right 1 unit to coordinate $-3$. Select the correct options.
(1) After rolling the die once, the probability that the piece is at distance 2 from the origin is $\frac { 1 } { 2 }$
(2) After rolling the die once, the expected value of the piece's coordinate is 0
(3) After rolling the die twice, the piece's coordinate could be $-5$
(4) After rolling the die twice, given that the sum of the two rolls is odd, the probability that the piece's coordinate is positive is $\frac { 4 } { 9 }$
(5) After rolling the die three times, the probability that the piece is at the origin is $\left( \frac { 1 } { 6 } \right) ^ { 3 }$