$X$ is a discrete random variable and $X\in \left\{0, 1, 2, 3\right\}.$ Given that $P\left(X>1\right)=0.24, P\left(0<\mathrm{X}<3\right)=0.5$ and $P(X=0$ or $2)=0.62,$ find $P(X⩽2\mid X>0).$

Four students are to be selected at random from a group that consists of seven boys and $x$ girls. The variables $B$ and $G$ are, respectively, the number of boys selected and the number of girls selected.

Given that $P\left(B=1\right)=P\left(B=2\right),$ find the value of $x.$

Four students are to be selected at random from a group that consists of seven boys and $x$ girls. The variables $B$ and $G$ are, respectively, the number of boys selected and the number of girls selected.

Given that $G\ne 3,$ find the probability that $G=4.$

A box contains $2$ green apples and $2$ red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable $X$ is the number of apples which have been taken when the process stops.

Show that $P\left(X=3\right)=\frac{1}{3}$

A box contains $2$ green apples and $2$ red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable $X$ is the number of apples which have been taken when the process stops.

Draw up the probability distribution table for $X.$ Another box contains $2$ yellow peppers and $5$ orange peppers. Three peppers are taken from the box without replacement.

A box contains $2$ green apples and $2$ red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable $X$ is the number of apples which have been taken when the process stops.

Given that at least $2$ of the peppers taken from the box are orange, find the probability that all $3$ peppers are orange.

In a particular discrete probability distribution the random variable $X$ takes the value $\frac{120}{r}$ with probability $\frac{r}{45},$ where $r$ takes all integer values from $1$ to $9$ inclusive.
Show that $P\left(X=40\right)=\frac{1}{15}.$

In a particular discrete probability distribution the random variable $X$ takes the value $\frac{120}{r}$ with probability $\frac{r}{45},$ where $r$ takes all integer values from $1$ to $9$ inclusive.
Construct the probability distribution table for $X.$

In a particular discrete probability distribution the random variable $X$ takes the value $\frac{120}{r}$ with probability $\frac{r}{45},$ where $r$ takes all integer values from $1$ to $9$ inclusive.

Which is the modal value of $X$ ?