Probability Distributions (AQA A Level Maths: Statistics): Exam Questions

The table shows the probability distribution of the number of previous owners, $N$, for a sample of cars taken from the Large Data Set.

Find the value of $\text{P}(1\le N<5)$

A random variable, $X$ , is defined as the number of heads when the three coins are tossed.

Given that for each coin the probability of getting heads is  $\frac{2}{3}$ , complete the following probability distribution table for $X$.

A spinner has four sections labelled 1, 3, 5 and 7. The spinner is spun and the number it lands on is represented by the random variable $X$ which has the probability function

$\mathrm{P}\left(X=x\right)=\{\begin{array}{l}kxx=1,3,5,7\\ 0\mathrm{otherwise}\end{array}$

 Find the value of $k$.

The spinner is spun twice. The random variable $Y$ represents the number of times that the spinner lands on the section labelled 3.

Complete the probability distribution of $Y$.

The random variable $X$ has the probability function

$\mathrm{P}\left(X=x\right)=\{\begin{array}{l}0.23x=-1,4\\ kx=0,2\\ 0.13x=1,3\\ 0\mathrm{otherwise}\end{array}$

Find the value of $k$.

Find $\mathrm{P}\left(X^2<3\right)$.

A discrete random variable $X$ has the probability distribution

Given that $\mathrm{P}\left(X=0\right)=\mathrm{P}\left(X>2\right)$, find the value of $p$ and the value of $q$.

A student claims that a random variable $X$ has a probability distribution defined by the following function:

$\mathrm{P}\left(X=x\right)=\{\begin{array}{l}\frac{x^2}{30}x=-1,1,3,5\\ 0\mathrm{otherwise}\end{array}$

Explain how you know that the student’s function does not describe a probability distribution.

Given that the correct probability mass function is of the form

$\mathrm{P}\left(X=x\right)=\{\begin{array}{l}\frac{x^2}{k}x=-1,1,3,5\\ 0\mathrm{otherwise}\end{array}$

where $k$ is a constant, write down the value of $k$.

The student claims that another random variable $Y$ can be defined by the following function:

$\mathrm{P}\left(Y=y\right)=\{\begin{array}{l}\frac{y^3}{152}y=-1,1,3,5\\ 0\mathrm{otherwise}\end{array}$

State, with a reason, whether the function is a valid probability mass function.

The probability that a biased coin lands on heads when flipped is $\frac{3}{5}$.

The coin is flipped four times. The random variable $H$ represents the number of times it lands on heads and the random variable $T$represents the number of times it lands on tails.

The random variable $X$ is defined as the non-negative difference between $H$ and $T$.

Complete the probability distribution for $X$. You may not need to use all the columns.

The random variable $X$ has the probability function

$\mathrm{P}\left(X=x\right)=\{\begin{array}{l}0.21x=1,2\\ \left(9-x\right)kx=3,6\\ 0.11x=4,5\\ 0\mathrm{otherwise}\end{array}$

Find the value of $k$.

A video game contains six levels. The number of levels that a player successfully completes is modelled by the random variable $X$ defined in part (a).

Two players each play the video game. Find the probability that, between them, they complete exactly 10 levels.

State an assumption that you made in part (b).

A discrete random variable $X$ has the probability distribution shown in the following table.

Find the value of $p$.

The random variables $X_1$ and $X_2$ are independent and each have the same distribution as $X$.

The random variable $Y$ is defined as $Y=X_1\times X_2$, the product of $X_1$ and $X_2$.

Fully describe the probability distribution for $Y$.

Leonidas is playing a game with a fair six-sided dice on which the faces are numbered 1 to 6.  He rolls the dice until either it lands on a 6 or he has rolled the dice four times.  The random variable $X$ is defined as the number of times that the dice is rolled.

Complete the probability distribution of $X$.

Find $\mathrm{P}\left(X^2\ge 5X-5\right)$.

Two biased coins are tossed. For each coin, the probability of getting heads is  $\frac{1}{3}$. The number of heads is represented by the random variable $H$.

A fair spinner with three sectors numbered 1 to 3 is spun. The number it lands on is represented by the random variable $S$.

The random variable, $X$, is defined as the product of the number of heads and the number on the spinner, such that $X=H\times S$.

Complete the following probability distribution for $X$.

The random variable $X$ has the probability function

$\mathrm{P}\left(X=x\right)=\frac{x^2}{495},x=p,2p,3p,4p,5p$

where  $p>0$is a constant.

Find $\mathrm{P}(3<X\le 12)$.

The independent random variables $X$  and $Y$  are defined by the probability distributions

$\mathrm{P}(X=x)=p,x=1,2,3,5,8,11$

$\mathrm{P}(Y=y)=\frac{q}{y},y=1,3,6$

where $p$ and $q$ are constants.

 Find  $\mathrm{P}(X>Y)$.

Leofranc is playing a gambling game with a fair six-sided dice on which the faces are numbered 1 to 6. He chooses a ‘lucky number’ between 1 and 6, and rolls the dice until either his lucky number appears or he has rolled the dice four times. 

• If his lucky number appears on the first roll, he receives £5.

• If his lucky number appears on the second roll, he receives £3.

• If his lucky number appears on the third roll, he receives £2.

• If his lucky number appears on the fourth roll, he receives £1.

• If his lucky number has not appeared by the fourth roll, he receives nothing.

The random variable $W$ is defined to be the amount of money, in pounds, that Leofranc receives.

Fully describe the probability distribution of $W$.

Each game costs Leofranc £2 to play.

Leofranc plays the game once. Find the probability that Leofranc receives at least his money back from the game.