Two fair $4$-sided dice, both with faces marked $1, 2, 3$ and $4,$ are rolled.
Event $A$ is 'the sum of the numbers obtained is a prime number'.
Event $B$ is 'the product of the numbers obtained is an even number'

Determine, with justification, whether events $A$ and $B$ are independent.

Two fair $4$-sided dice, both with faces marked $1, 2, 3$ and $4,$ are rolled.
Event $A$ is 'the sum of the numbers obtained is a prime number'.
Event $B$ is 'the product of the numbers obtained is an even number'

Give a reason why events $A$ and $B$ are not mutually exclusive.

Two ordinary fair dice are rolled.
Event $X$ is 'the product of the two numbers obtained is odd'.
Event $Y$ is 'the sum of the two numbers obtained is a multiple of $3 \text{'}.$

Determine, giving reasons for your answer, whether $X$ and $Y$ are independent.

Two ordinary fair dice are rolled.
Event $X$ is 'the product of the two numbers obtained is odd'.
Event $Y$ is 'the sum of the two numbers obtained is a multiple of $3 \text{'}.$

Are events $X$ and $Y$ mutually exclusive? Justify your answer.

Two hundred children are categorised by gender and by whether or not they own a bicycle. Of the $108$ males, $60$ own a bicycle, and altogether $90$ children do not own a bicycle.

Tabulate these data.

Two hundred children are categorised by gender and by whether or not they own a bicycle. Of the $108$ males, $60$ own a bicycle, and altogether $90$ children do not own a bicycle.

Determine, giving reasons for your answer, whether ownership of a bicycle is independent of gender for these $200$ children.

Two hundred children are categorised by gender and by whether or not they own a bicycle. Of the $108$ males, $60$ own a bicycle, and altogether $90$ children do not own a bicycle.

What percentage of the females and what percentage of the males own bicycles?

At an election, it was found that people voted for Party $X$ independently of their income group. The following table shows that $12400$ people from three income groups voted altogether, and that $7440$ of them voted for Party $X$.

Find the value of $a$, of $b$ and of $c$.