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

Julia selects 3 letters at random, one at a time without replacement, from the word

V A R I A N C E

The discrete random variable $X$ represents the number of times she selects a letter A.

Find the complete probability distribution of $X$.

Helen believes that the random variable $C$, representing cloud cover from the large data set, can be modelled by a discrete uniform distribution.

Write down the probability distribution for $C$.

Using this model, find the probability that cloud cover is less than 50%.

Helen used all the data from the large data set for Hurn in 2015 and found that the proportion of days with cloud cover of less than 50% was 0.315

Comment on the suitability of Helen’s model in the light of this information.

Suggest an appropriate refinement to Helen’s model.

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 $X$.

The random variable $X$  has the probability function

$\mathrm{P}\left(X=x\right)=\{\begin{array}{l}\frac{1}{k}x=1,2,3,4,5\\ 0\mathrm{otherwise}\end{array}$

(i) Write down the value of $k$.

(ii) Write down the name of this probability distribution.

$X_1$ and $X_2$ are independent observations of the random variable $X$.

Find $\mathrm{P}\left(X_1<X_2\right)$.

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 discrete random variable $X$ has the following probability distribution

where

• $a$, $b$ and $c$ are distinct integers ($a<b<c$)

• all the probabilities are greater than zero

Find

(i) the value of $a$

(ii) the value of $b$

(iii) the value of $c$

Show your working clearly.

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

Find $\mathrm{P}\left(X_1=X_2\right)$.

The discrete random variable $D$ has the following probability distribution

where $k$ is a constant.

Show that the value of $k$ is $\frac{600}{137}$.

The random variables $D_1$ and $D_2$ are independent and each have the same distribution as $D$.

Find $\mathrm{P}(D1+D2=80)$.

Give your answer to 3 significant figures.

A single observation of $D$ is made.

The value obtained, $d$, is the common difference of an arithmetic sequence.

The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral $Q$.

Find the exact probability that the smallest angle of $Q$ is more than 50°.

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.

Tisam is playing a game.

She uses a ball, a cup and a spinner.

The random variable $X$ represents the number the spinner lands on when it is spun.

The probability distribution of $X$ is given in the following table

where $a$, $b$, $c$ and $d$ are probabilities.

To play the game

• the spinner is spun to obtain a value of $x$

• Tisam then stands $x$ cm from the cup and tries to throw the ball into the cup

The event $S$ represents the event that Tisam successfully throws the ball into the cup.

To model this game Tisam assumes that

• $\mathrm{P}\left(S|\left\{X=x\right\}\right)=\frac{k}{x}$ where $k$ is a constant

• $\mathrm{P}\left(S\cap \left\{X=x\right\}\right)$ should be the same whatever value of $x$ is obtained from the spinner

Using Tisam’s model, show that $c=\frac{8}{5}b$.

Using Tisam’s model, find the probability distribution of $X$.

Nav tries, a large number of times, to throw the ball into the cup from a distance of 100 cm.

He successfully gets the ball in the cup 30% of the time.

State, giving a reason, why Tisam’s model of this game is not suitable to describe Nav playing the game for all values of $X$.

Manon has two biased spinners, one red and one green.

The random variable $R$ represents the score when the red spinner is spun.

The random variable $G$ represents the score when the green spinner is spun.

The probability distributions for $R$ and $G$ are given below.

Manon spins each spinner once and adds the two scores.

Find the probability that

(i) the sum of the two scores is 7

(ii) the sum of the two scores is less than 4

The random variable $X=mR+nG$ where $m$ and $n$ are integers.

$\mathrm{P}(X=20)=\frac{1}{6}$and$\mathrm{P}\left(X=50\right)=\frac{1}{4}$

Find the value of $m$ and the value of $n$.