Each child in a group of $80$ was asked whether they regularly read $\left(R\right)$ or regularly watch a movie $\left(M\right)$ The results are given in the Venn diagram opposite. One child is selected at random from the group. Event $R$ is 'a child who regularly reads is selected' and event $M$ is 'a child who regularly watches a movie is selected'.

Determine, with justification, whether events $R$ and $M$ 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'

Find, in simplest form, the value of $\mathrm{P}\left(A\right)$, of $\mathrm{P}\left(B\right)$ and of $\mathrm{P}(A\cap B)$.

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.