Introduction to Probability & Statistics

Assignment 3, 2025/26

Instructions

Submit your answers to the four questions marked Hand-in. You should upload your solutions to the VLE as a single pdf file. Marks will be awarded for clear, logical explanations, as well as for correctness of solutions.
Solutions to questions marked have been released at the same time as this assignment, in case you want to check your answers or need a hint.
You should also look at the other questions in preparation for your Week 7 seminar.

Starters

These questions should help you to gain confidence with the basics.

S1. Each morning a student rolls a die and starts studying if she gets a 6. Otherwise, she stays in bed. However, during the four months of exams, she tosses a coin instead of rolling a die and studies if she gets a Head. On a (randomly chosen) morning, the student is studying. What is the probability that it is exam period? (Assume for simplicity that each month has the same number of days!)

Answer

Let $W=\{\text{student is working}\}$ and} $E=\{\text{exam period}\}$. Then Bayes’ theorem tells us that
\[ \begin{split} \mathbb{P}\left(E\,|\,W\right)&=\frac{\mathbb{P}\left(W\,|\,E\right)\mathbb{P}\left(E\right)}{\mathbb{P}\left(W\,|\,E\right)\mathbb{P}\left(E\right)+\mathbb{P}\left(W\,|\,E^c\right)\mathbb{P}\left(E^c\right)}\\ &=\frac{\dfrac{1}{2}\cdot\dfrac{1}{3}}{\dfrac{1}{2}\cdot\dfrac{1}{3}+\dfrac{1}{6}\cdot\dfrac{2}{3}} =\frac{3}{5}. \end{split} \]

S2. Consider a discrete random variable $V$ taking values in $\{1,2,3,4\}$ with mass function $p_V$: \[ p_V(x) = \begin{cases} xc^x & \quad\text{if $x=1,2$} \\ c(5-x) & \quad\text{if $x=3,4$}. \end{cases} \] What value of $c$ makes $p_V$ a mass function?

Answer

To be a mass function, we need $p_V(x)\geq 0$ for all $x$ and $\sum_x p_V(x) = 1$. For the second of these constraints we need $c + 2c^2 + 2c + c = 1$, and so $c = -1 \pm \sqrt{6}/2$. But if we take $c=-1-\sqrt{6}/2$ then we get (for example) $p_V(3)<0$, which isn’t allowed. Therefore, we have to take $c=-1+\sqrt{6}/2>0$.

S3. Hand-in
Consider a discrete random variable $Y$ taking values in $\{1,2,3,4,5\}$ with mass function $p_Y$ given by \[ p_Y(y) = \begin{cases} c y^2 & \quad\text{if $y=1,2$} \\ (6-y)/8 & \quad\text{if $y=3,4,5$}. \end{cases} \] What value of $c$ makes $p_Y$ a mass function?

S4. What is the probability that exactly 3 heads are obtained in 5 tosses of a fair coin?

S5. A fair die is thrown until the sum of the results of the throws exceeds 6. The random variable $X$ is the number of throws needed for this. Let $F_X$ be the distribution function of $X$. Determine $F_X(1)$, $F_X(2)$ and $F_X(7)$.

Answer

There is no possibility to exceed $6$ already on the first throw because the die only has numbers up to $6$. So $F_X(1)=\mathbb{P}\left(X\leq 1\right)=0.$

There are, however, $21$ outcomes of the throw of two dice that give a sum greater than $6$. There are $36$ outcomes altogether, all equally likely, so \[ F_X(2)=\mathbb{P}\left(X\leq 2\right)=\mathbb{P}\left(X= 2\right)=\frac{21}{36}=\frac{7}{12}\,. \] Finally, there is no way to avoid getting a sum of more than six once one has more than six throws, because each die will show at least one. So $F_X(7)=\mathbb{P}\left(X\leq 7\right)=1-\mathbb{P}\left(X>7\right)=1-0=1$.

S6. The probability density function $f_X$ of a continuous random variable $X$ is given by \[ f_X(x)=\begin{cases} x(3-x)+c &\text{ if $1\leq x\leq2$}\\ 0&\text{ otherwise\,. }\end{cases} \] Compute $c$. Work out the distribution function of $X$.

Answer

The constant $c$ is determined by the requirement that the integral of the density function over the entire real line is equal to one: \[ 1=\int_{-\infty}^\infty f_X(x)dx=\int_1^2 (x(3-x)+c) dx = 13/6+c\,. \] This implies that $c=-7/6$. Note that for this value of $c$ the function $f_X$ is non-negative everywhere, which is also required in order for it to be a density.

The distribution function is obtained as an integral over the density function, but not over the whole real line but only from $-\infty$ up to $x$: \[ F_X(x)=\begin{cases} 0 & \text{ if $x<1$}\\ \int_1^x f_X(s)ds=\int_1^x (s(3-s)-7/6)ds= \frac{9x^2-2x^3-7x}{6} &\text{ if $1\leq x\leq2$}\\ 1 & \text{if $2< x$.} \end{cases} \]

S7. Hand-in
The probability density function $f_X$ of a continuous random variable $X$ is given by \[ f_X(x)=\begin{cases} c x^2 &\text{ if $0\leq x\leq 1$}\\ 1/2 & \text{if $2\leq x\leq 3$}\\ 0&\text{ otherwise\,. }\end{cases} \] Compute $c$, and hence calculate $\mathbb{P}\left(X> 1/2\right)$.

S8. Let $Y\sim\mbox{\textup{Poisson}}(2)$. Write down the mass function for $Y$. What is $\mathbb{P}\left(Y\geq 2\right)$?

S9. A continuous random variable $X$ has a distribution function $F$ which satisfies \[ F(x) = \begin{cases}0 &\text{ for } x < 0\\\frac{4x^2+3x}{7} &\text{ for } 0 \leq x \leq 1\\ 1 &\text{ for } x > 1\,. \end{cases} \] Determine $\mathbb{E}\left[X\right]$.

Mains

These are important, and cover some of the most substantial parts of the course.

M1. Hand-in
Consider the probability space corresponding to throwing a fair die. Give an example of a random variable $X$ whose image is $\{1, 2, 3\}$ and for which $F_X(2.7)=5/6$.

M2 Hand-in
A box contains four coins: one has a Head on both sides; one has a Tail on both sides; the other two coins are normal and fair. A coin is chosen at random from the box and tossed three times. What is the probability that the two-Tailed coin was chosen, given that all three tosses are Tails?

M3. Let $\lambda>0$ be some positive number. Find $c$ such that the following function is a density function: \[ f(x) =ce^{-\lambda \left\vert x-1\right\vert }\,, \quad x\in \mathbb{R}. \]

Answer

For $f$ to be a density, it must be non-negative on $\mathbb{R}$ and satisfy $\int_{-\infty}^\infty f(x) dx = 1$. Since the exponential function is non-negative, $f$ is non-negative everywhere as long as $c\geq 0$

Note that \[ \begin{split} \int_{-\infty}^{\infty}e^{-\lambda |x-1|}dx &= \int_{-\infty}^1e^{-\lambda (1-x)}dx+\int_1^{\infty}e^{-\lambda (x-1)}dx\\ &=\frac{1}{\lambda}\left[e^{-\lambda(1- x)}\right]_{-\infty}^1-\frac{1}{\lambda}\left[e^{-\lambda (x-1)}\right]_1^{\infty}=\frac{2}{\lambda}\,. \end{split} \] So $f(\cdot)$ is a density if and only if $c=\lambda/2$.

M4. Recall the Chevalier de Méré’s problem: two dice are thrown 24 times – you win £1 if at least one double 6 is thrown, otherwise you lose £1. Show that you expect to lose money on average if you play this game.

Answer

The probability of throwing a double 6 with one throw of two dice is $1/36$. The probability of there being no occurrences of a double 6 in 24 throws of two dice is therefore $(35/36)^{24} \approx 0.509$. Letting $X$ denote your winnings from the game, this is a random variable satisfying \[ \mathbb{P}\left(X=1\right) \approx 0.491\,, \quad \text{and} \quad \mathbb{P}\left(X=-1\right) \approx 0.509 \,. \]

Your expected winnings are hence given by \[ \mathbb{E}\left[X\right] \approx 0.491 - 0.509\approx -0.018 \ <0 \,. \] Therefore you expect to lose on average in this game.

M5. Suppose that $A$ and $B$ are independent events. Show that $A^c$ and $B^c$ are independent.

M6. Consider a roulette wheel, with numbers 00, 0, 1, 2, 3, …, 36 (38 numbers in total). A ball is thrown onto the wheel as it is spinning, and comes to rest by one of the numbers. You always bet that the ball will stop on one of the numbers 1, 2, …, 12. Let $N$ be the random variable giving the number of bets that you lose before your first win. Calculate $p_N(0)$, $p_N(5)$ and $F_N(5)$.

M7. Consider a discrete random variable $Y$ taking values in the set $\{0,1,\dots,8\}$ and with probability mass function $p_Y$ given by \[ p_Y(k) = \begin{cases} c\,a & \quad\text{if $k=0,1,2,3,4,5$} \\ c\,a^2 & \quad\text{if $k=6,7$} \\ c(1-a)^2 & \quad\text{if $k=8$,} \end{cases} \] where $a$ is some fixed number between 0 and 1 and $c$ is a constant to be determined. What value of $c$ makes $p_Y$ a mass function? (The answer is a function of $a$.)

M8. Let $X\sim\mbox{\textup{Bern}}(p)$. Let $Y=1-X$ and $V=X^2$. Show that

$Y\sim\mbox{\textup{Bern}}(1-p)$;
$V\sim\mbox{\textup{Bern}}(p)$.

M9. $N\geq 3$ people go for coffee. Each person flips a fair coin: if all but one of the coins shows the same face (Head or Tail), then the odd person out pays for all the drinks; if not then the coins are tossed again until this event occurs. How many times on average must each person toss their coin before somebody is selected in this way?

M10. Prove the memoryless property of the exponential distribution which states that if $X\sim\mbox{\textup{Exp}}(\lambda)$ then for any $t,s\geq 0$, \[ \mathbb{P}\left(X>t+s\,|\,X>s\right)=\mathbb{P}\left(X>t\right). \]

Desserts

Still hungry for more? Try these if you want to push yourself further. (These are mostly harder than I’d expect you to answer in an exam, or involve non-examinable material.)

D1. $N\ge 3$ people go for coffee. Each person flips a coin, of which $N-1$ are fair, and one has probability $q$ of coming up Heads (for some $q\in[0,1]$): if all but one of the coins shows the same face (Head or Tail), then the odd person out pays for all the drinks; if not then the coins are tossed again until this event occurs. How does the expected number of coin tosses each person makes vary with $q$?

D2. Alice and Bob decide to duel, using just one six-shot revolver, and one bullet, between them. They decide to duel in the following way: with the bullet inserted into the revolver, Alice will spin the cylinder and shoot at Bob (killing him if the gun fires); if Alice misses, Bob will spin the cylinder and shoot at Alice. Assuming there is a $1/6$ probability that the revolver will fire each time (since the revolver has 6 chambers), what is

the distribution of the number of turns $T$ until the gun fires?
the probability that Alice wins the duel?

Challenge question

At time 0, a bag contains 1 Red and 1 Black ball. Just before each time $n=1,2,3,\dots$, a ball is chosen at random from the bag and then replaced, along with a new ball of the same colour. For $n=0,1,2,\dots$ and $1\leq r\leq n+1$, let \[ p_{n,r} = \mathbb{P}\left(\text{at time $n$, the bag contains exactly $r$ Red balls}\right) \,. \]

Write down the values of $p_{0,1}$, $p_{1,1}$ and $p_{1,2}$;
By conditioning on the possible numbers of red balls at time 1, calculate $p_{2,r}$ for $r=1,2,3$;
Can you find (and prove!) a general formula for $p_{n,r}$? [HINT: guess at the right answer using your findings from parts (a) and (b). (Calculate $p_{3,r}$ explicitly if you can’t see a pattern.) Then use induction.]