A fair six-sided dice has a "$1$" on one face a ''$2$'' on two of its laces and a "$3$" on the remaining three faces.

The dice is thrown twice, and $T$ is the random variable "the sum of scores from both throws''.

Find the probability that the total score is more than $4$.

A board game is played by moving a counter $S$ spaces forward at a time, where $S$ is determined by the following rule:

A fair six-sided dice is thrown once. $S$ half the number shown on the dice if that number is even; otherwise $S$ is twice the number shown on the dice. Write out a table showing the possible value of $S$ and their probabilities.

A board game is played by moving a counter $S$ spaces forward at a time, where $S$ is determined by the following rule: A fair six-sided dice is thrown once. $S$ half the number shown on the dice if that number is even; otherwise $S$ is twice the number shown on the dice. Find the probability that, in a single turn, a player moves their counter forward more than $2$ spaces.

The random variable $X$ has the probability distribution shown.

Find the value of $c$.

The discrete random variable $X$ can take only the values $0, 1, 2, 3, 4, 5$. The probability distribution of $X$ is given by the following 

$P(X=0)=P(X=1)=P(X= 2) = a$

$P(X=3)=P(X=4)=P(X= 5) = b$

$P(X\ge 2)=3P(X<2)$

where $a$ and $b$ are constants.

Determine the values of $a$ and $b$.

The discrete random variable $X$ can take only the values $0, 1, 2, 3, 4, 5$. The probability distribution of $X$ is given by the following 

$P(X=0)=P(X=1)=P(X= 2) = a$

$P(X=3)=P(X=4)=P(X= 5) = b$

$P(X\ge 2)=3P(X<2)$

where $a$ and $b$ are constants.

Determine the probability that the sum of two independent observations from this distribution exceeds $7$.

The discrete random variables A and B are independent and have the following distributions.

The random variable $C$ is the sum of one observation from $A$ and one observation from $B$.

Show that $P(C=3)= \frac{5}{18}$.

The discrete random variables A and B are independent and have the following distributions.

The random variable $C$ is the sum of one observation from A and one observation from $B$. Tabulate the probability distribution for $C$.

Complete this probability distribution, in terms of $k$, for a discrete random variable $X$

What range of values can $k$ take? Give your answer in the form $a\le k\le b, a, b\in Q$