Let $X, N, Y$ and $Z$ be random variables. The variables $X$ and $N$ are independent of each other. $X$ is uniformly distributed between -1 and $1 ; N$ follows Normal distribution with zero mean and unity variance.

$Y$ and $Z$ are defined as, $Y=X+N$ and $Z=X^2+N$.

Which of the following pairs represents the values of correlation between $X$ and $Y$ and that between $X$ and $Z$ ?

Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained. The expectation of $X$ is ___________ (rounded off to two decimal places).

In a high school having equal number of boy students and girl students, 75% of the students study Science and the remaining 25% students study Commerce. Commerce students are two times more likely to be a boy than are Science students. The amount of information gained in knowing that a randomly selected girl student studies Commerce (rounded off to three decimal place) is ______ bits.

A box contains the following three coins.

I. A fair coin head on one face and tail on the other face.

II. A coin with heads to both the faces.

III. A coin with tails on both the faces.

A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the probability of getting a head in the second toss is