Let $k$ be an integer greater than or equal to 2. We say that random variables $(Y_n)_{n \geq 1}$ are $k$-independent if $$\mathbb{E}\left[\psi_1(Y_{n_1}) \cdots \psi_k(Y_{n_k})\right] = \mathbb{E}\left[\psi_1(Y_{n_1})\right] \cdots \mathbb{E}\left[\psi_k(Y_{n_k})\right]$$ for all indices $1 \leq n_1 < \cdots < n_k$ and for all bounded functions $\psi_1, \ldots, \psi_k : \mathbb{R} \rightarrow \mathbb{R}$.
I.4.a) Prove that $k$-independence implies $j$-independence if $j \leq k$.
I.4.b) What is $N$-independence for $N$ random variables $Y_1, \ldots, Y_N$?
I.4.c) Let $Y_1$ and $Y_2$ be independent random variables with uniform distribution on $\{0,1\}$: for $n = 1,2$, $$\mathbb{P}(Y_n = 0) = \mathbb{P}(Y_n = 1) = \frac{1}{2}$$ Let $Y_3$ be the random variable on $\{0,1\}$ defined by $$Y_3 := Y_1 + Y_2 \quad \bmod 2$$ Prove that the random variables $(Y_1, Y_2, Y_3)$ are 2-independent but not 3-independent.

An experiment has 10 equally likely outcomes. Let $A$ and $B$ be two non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is
(A) 2, 4 or 8
(B) 3, 6 or 9
(C) 4 or 8
(D) 5 or 10

Of the three independent events $E _ { 1 } , E _ { 2 }$ and $E _ { 3 }$, the probability that only $E _ { 1 }$ occurs is $\alpha$, only $E _ { 2 }$ occurs is $\beta$ and only $E _ { 3 }$ occurs is $\gamma$. Let the probability $p$ that none of events $E _ { 1 } , E _ { 2 }$ or $E _ { 3 }$ occurs satisfy the equations $( \alpha - 2 \beta ) p = \alpha \beta$ and $( \beta - 3 \gamma ) p = 2 \beta \gamma$. All the given probabilities are assumed to lie in the interval $( 0,1 )$.
$$\text { Then } \frac { \text { Probability of occurrence of } E _ { 1 } } { \text { Probability of occurrence of } E _ { 3 } } =$$

Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2 , respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is
(1) 0.06
(2) 0.14
(3) 0.2
(4) None of these

Let $A$ and $B$ be two events such that $P ( \overline { A \cup B } ) = \frac { 1 } { 6 } , P ( A \cap B ) = \frac { 1 } { 4 }$ and $P ( \bar { A } ) = \frac { 1 } { 4 }$, where $\bar { A }$ stands for the complement of the event $A$. Then the events $A$ and $B$ are
(1) Independent but not equally likely.
(2) Independent and equally likely.
(3) Mutually exclusive and independent.
(4) Equally likely but not independent.

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2$ is the event that die $B$ shows up two and $E_3$ is the event that the sum of dice $A$ and $B$ is odd, then which of the following statements is NOT true? (1) $E_1$ and $E_3$ are independent. (2) $E_1$, $E_2$ and $E_3$ are independent. (3) $E_1$ and $E_2$ are independent. (4) $E_2$ and $E_3$ are independent.

Three persons $\mathrm { P } , \mathrm { Q }$ and R independently try to hit a target. If the probabilities of their hitting the target are $\frac { 3 } { 4 } , \frac { 1 } { 2 }$ and $\frac { 5 } { 8 }$ respectively, then the probability that the target is hit by P or Q but not by R is:
(1) $\frac { 39 } { 64 }$
(2) $\frac { 21 } { 64 }$
(3) $\frac { 9 } { 64 }$
(4) $\frac { 15 } { 64 }$

Let $A = \{ 0 , 3 , 4 , 6 , 7 , 8 , 9 , 10 \}$ and $R$ be the relation defined on $A$ such that $R = \{ ( x , y ) \in A \times A : x - y$ is odd positive integer or $x - y = 2 \}$. The minimum number of elements that must be added to the relation $R$, so that it is a symmetric relation, is equal to $\_\_\_\_$

Let a relation R on $\mathrm { N } \times N$ be defined as: $\left( x _ { 1 } , y _ { 1 } \right) \mathrm { R } \left( x _ { 2 } , y _ { 2 } \right)$ if and only if $x _ { 1 } \leq x _ { 2 }$ or $y _ { 1 } \leq y _ { 2 }$. Consider the two statements: (I) R is reflexive but not symmetric. (II) $R$ is transitive Then which one of the following is true?
(1) Both (I) and (II) are correct.
(2) Only (II) is correct.
(3) Neither (I) nor (II) is correct.
(4) Only (I) is correct.

An opaque bag contains blue and green balls, each marked with a number 1 or 2. The quantities are shown in the table below. For example, there are 2 blue balls marked with number 1.

	Blue	Green
Number 1	2	4
Number 2	3	$k$

A ball is randomly drawn from the bag (each ball has an equal probability of being drawn). Given that the event of drawing a blue ball and the event of drawing a ball marked with 1 are independent, what is the value of $k$?
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6