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Let \( N \) denote the number that turns up when a fair die is rolled. If the probability that the system of equations
\( x+y+z=1 \)
\( 2x+Ny+2z=2 \)
\( 3x+3y+Nz=3 \)
has unique solution is \( \dfrac{k}{6} \), then the sum of value of \( k \) and all possible values of \( N \) is:

A coin is biased so that the head is \( 3 \) times as likely to occur as tail. This coin is tossed until a head or three tails occur. If \( X \) denotes the number of tosses of the coin, then the mean of \( X \) is:

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is \( p \). Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is \( q \). If \( p : q = m : n \), where \( m \) and \( n \) are coprime, then \( m + n \) is equal to ________.

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable \( X \) denote the number of rotten apples. If \( \mu \) and \( \sigma^2 \) represent mean and variance of \( X \), respectively, then \( 10(\mu^2+\sigma^2) \) is equal to:

Let $S=\{w_1,w_2,\dots\}$ be the sample space associated to a random experiment. Let $P(w_n)=\frac{P(w_{n-1})}{2},n\geq2$. Let $A=\{2k+3l:k,l\in\mathbb{N}\}$ and $B=\{w_n:n\in A\}$. Then $P(B)$ is equal to :

In a binomial distribution \( B(n,p) \), the sum and the product of the mean and the variance are \( 5 \) and \( 6 \) respectively, then \( 6(n+p-q) \) is equal to:

Let \( N \) denote the sum of the numbers obtained when two dice are rolled. If the probability that \( 2^N < N! \) is \( \dfrac{m}{n} \), where \( m \) and \( n \) are coprime, then \( 4m - 3n \) is equal to:

A bag contains \( 6 \) balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least \( 5 \) black balls is:

If an unbiased die, marked with \(-2, -1, 0, 1, 2, 3\) on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is:

Let $\Omega$ be the sample space and $A \subseteq \Omega$ be an event. 
Given below are two statements:
Statement I :If $\text{P}(A) = 0$, then $A = \phi$
Statement II : If $\text{P}(A) = 1$, then $A = \Omega$
Then

Fifteen football players of a club-team are given \( 15 \) T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least \( 3 \) players pick the correct T-shirt is:

Let a die be rolled \( n \) times. Let the probability of getting odd numbers seven times be equal to the probability of getting even numbers twice is \( \dfrac{k}{2^{15}} \), then \( k \) is equal to:

Two dice are thrown independently. Let \( A \) be the event that the number appeared on the \( 1^{\text{st}} \) die is less than the number appeared on the \( 2^{\text{nd}} \) die, \( B \) be the event that the number appeared on the \( 1^{\text{st}} \) die is even and that on the second die is odd, and \( C \) be the event that the number appeared on the \( 1^{\text{st}} \) die is odd and that on the \( 2^{\text{nd}} \) is even. Then:

\( 25\% \) of the population are smokers. A smoker has \( 27 \) times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is \( \frac{k}{10} \). Then the value of \( k \) is ____.

If the probability that the random variable \( X \) takes values \( x \) is given by \( P(X=x)=k(x+1)3^{-x},\; x=0,1,2,3,\ldots \), where \( k \) is a constant, then \( P(X \ge 2) \) is equal to:

A pair of dice is thrown \( 5 \) times. For each throw, a total of \( 5 \) is considered a success. If the probability of at least \( 4 \) successes is \( \dfrac{k}{3^{11}} \), then \( k \) is equal to:

Let the probability of getting head for a biased coin be \( \dfrac{1}{4} \). It is tossed repeatedly until a head appears. Let \( N \) be the number of tosses required. If the probability that the equation \( 64x^2 + 5Nx + 1 = 0 \) has no real root is \( \dfrac{p}{q} \), where \( p \) and \( q \) are coprime, then \( q - p \) is equal to _______.

Three dice are rolled. If the probability of getting different numbers on the three dice is \( \dfrac{p}{q} \), where \( p \) and \( q \) are co-prime, then \( q - p \) is equal to:

A fair \( n\ (n>1) \) faces die is rolled repeatedly until a number less than \( n \) appears. If the mean of the number of tosses required is \( \dfrac{n}{9} \), then \( n \) is equal to_____.

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

In a bolt factory, machines \( A \), \( B \) and \( C \) manufacture respectively \( 20\% \), \( 30\% \) and \( 50\% \) of the total bolts. Of their output \( 3 \), \( 4 \) and \( 2 \) percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine \( C \) is:

Three urns \( A \), \( B \) and \( C \) contain \( 4 \) red, \( 6 \) black; \( 5 \) red, \( 5 \) black; and \( \lambda \) red, \( 4 \) black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that is drawn from urn \( C \) is \( 0.4 \) then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola \( y^2=\lambda x \) with one vertex at the vertex of the parabola, is:

Let $M$ be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $S = \left\{ x \in \mathbb{Z} : x(66-x) \geq \frac{5}{9}M \right\}$ and the event $A = \{ x \in S : x \text{ is a multiple of } 3 \}$. Then $P(A)$ is equal to:

A bag contains \( 6 \) white and \( 4 \) black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is: