Exercises sheet 1

Counting (Combinatorics)

Exercice 1 Separated questions:

In computing, the binary system is used to code characters. A bit (binary digit) is an element that takes the value 0 or the value 1. With 8 binary digits (one byte), how many characters can we code?

In a sports competition grouping 18 athletes, one gold, one silver and one bronze medal are awarded. How many distributions are there possible (before the competition, of course…)?

A sports tournament has 8 participating teams. Each team must meet all the others once. How many games must we organize?

Exercice 2 A die 🎲 is rolled 3 times in a row, and we are interested in the total of the points obtained. In how many ways can we get:

a total of 16.
a total of 15.
at least 15.

Exercice 3 In a company, there are 12 singles among the 30 employees. We wish to make a survey: for that we choose a sample of four people in this department.

How many different samples can we obtain?
What is the number of samples containing no single person?
What is the number of samples containing at least one single?

Exercice 4 Five cards are drawn simultaneously from a deck of 52 cards.

How many different draws can we get?
How many different draws can we get? containing:
1. 5 diamonds or 5 hearts;
2. 2 hearts and 3 spades;
3. at least 1 king
4. at most 1 king.

Events

Exercice 5 Let $A$,$B$ and $C$ be three events of a measurable space $(\Omega,\mathcal{A})$. Express in terms of $A$,$B$ and $C$ and the operations (Union, intersection and complementary) the following events below:

$A$ alone (among the 3 events) occurs.
$A$ and $C$ occur, but not $B$.
All three events occur.
At least one of the 3 events occurs.
None of the three events occur.

Probability

Exercice 6 Let the events $A$ et $B$ such that $P(A) = x, P(B) = y$ and $P(A \cup B) = z$. Find the following probabilities: $P(A \cap B), P(\bar{A} \cup \bar{B})$, and $P(A \cap \bar{B})$.

Exercice 7 Let $(\Omega,\mathcal{A},P)$ a probability space. Let $A$,$B$ et $C$ three events. Let: $E = A \cap \bar{B} \cap \bar{C}$ and $F = A \cap (B \cup C)$.

Show that $E$ and $F$ are mutually exclusive.
Show that $E \cup F = A$.
Knowing that $P(A)=0.6; P(A\cap B)=0.2; P(A\cap C)=0.1$ and $P(A\cap B \cap C)=0.05$: Compute $P(F)$ and $P(E)$.

Exercice 8

An urn contains 13 balls, 6 black, 3 white and 4 red. We draw 4 balls. Let

E: ” get exactly 2 white “.
F: ” get exactly 2 red “.

We suppose that we draw without replacement. Calculate the following probabilities: $P(E \cap F), P_F (E), P_E(F)$. Are the events E and F independent?
Repeat the exercise, assuming that the draw is made with replacement.

Exercice 9 An urn contains ten balls (6 white and 4 red). Two balls are drawn at random and successively from this urn. Calculate, in the case where the draw is made without replacement, then in the case where the draw is made with replacement, the following probabilities:

probability that the two balls are white.
probability that the two balls are of the same color.
probability that at least one of the balls drawn is white.

Bayes

Exercice 10

A new vaccine was tested on 12,500 people. 75 of them, including 35 pregnant women, had adverse reactions requiring hospitalization.

Knowing that this vaccine was administered to 680 pregnant women, what is the probability that a pregnant woman will have a secondary reaction if she receives the vaccine?
What is the probability that a non-pregnant person would have a secondary reaction?
A person had adverse reactions. What is the probability that this person is a pregnant woman?

Exercice 11

To go to high school, a student has the choice between 4 routes: A, B, C and D. The probability that he chooses A (respectively B, C) is $\frac{1}{3}$ (respectively $\frac{1}{4}$, $\frac{1}{12}$). The probability to arrive late by taking A (respectively B, C) is $\frac{1}{20}$ (respectively $\frac{1}{10}$, $\frac{1}{5}$). By borrowing D, he is never late.

What is the probability that the student chooses route D?
The student arrives late. What is the probability that he took route C?

Discrete random variables

Exercice 12 Let $X$ be a random variable which takes its values in the set $\{- 4, 2, 3, 4, 6, 7, 10\}$ and whose distribution is given by:

$x_i$	-4	2	3	4	6	7	10
$P(X=x_i)$	0.1	0.2	0.2	0.1	0.05	0.2	$k$

Calculate $k$.
Plot the law of $X$.
Calculate the following probabilities:

$P(X >3)$
$P(X \geq 3)$
$P(3 \leq X \leq 7)$
$P(3 < X < 9)$
$P(X+2 > 3)$
$P(X^2 > 4)$

Exercice 13 Two balls are chosen randomly from an urn containing 8 white balls, 4 black balls and 2 orange balls. Assume we receive 2 euros for each black ball drawn and we lose 1 euro for each white ball drawn. Let’s refer to the net gains by $X$.

What are the possible values for $X$ and the probabilities associated with these values ?
What is the expected value of $X$?

Exercice 14 An urn contains a ball with number 0, two with number 1 and four with number 3. Two balls are simultaneously extracted from this urn.

Determine the probability law of the random variable $X$ which represents the sum of the numbers obtained.
Determine the distribution function of $X$.
Calculate $E(X)$, $V(X)$ and $\sigma(X)$.

Exercice 15 et $X$ be a random variable that follows the uniform distribution (e.g. equiprobability of values of $X$) on the set $X(\Omega) = \{-3, -2, 1, 4\}$.

Give the law of $X$.
Calculate $E(X)$ and $V(X)$.

We define the random variable $Y=(X+1)^2$.
Give $Y(\Omega)$ and the law of $Y$.
Calculate $E(Y)$ using two different methods.

Exercice 16 Let $F$ be the distribution function of the random variable $X$ defined by:

\[F(x) = \left\{ \begin{array}{l l} 0 & \quad \text{si $x<0$}\\ 1/4 & \quad \text{si $0 \leq x < 1$}\\ 1/2 & \quad \text{si $1 \leq x < 4$}\\ c & \quad \text{si $x \geq 4$}\\ \end{array} \right.\]

Determine with justification the constant value $c$.
Calculate $P (1 \leq X < 5)$.
Calculate $P(X=1)+P(X=2)$.
Determine the probability distribution of $X$.

Exercice 17 Let X and Y be discrete random variables whose joint distribution is given by the following table:

$X$\$Y$	-1	0	2	5
0	0.10	0.05	0.15	0.05
1	0.15	0.20	0.25	0.05

What is the marginal distribution of X?
What is the marginal distribution of Y?
Calculate $P(Y \geq 0 / X = 1)$.
Compute $E(X)$, $E(Y)$, and $cov(X,Y)$.
Are the variables $X$ and $Y$ independent?

Exercice 18 Let $(X,Y)$ be a pair of random variables with values in $\mathbb{N}^2$ such that

\[\forall (p,q) \in \mathbb{N}^2, \quad P(X=p,Y=q) = \lambda \frac{p+q}{p! q! 2^{p+q}}\]

Determine $\lambda$.
Calculate the marginal distributions.
Are the variables $X$ and $Y$ independent?

Common discrete distributions

Exercice 19 An urn contains 2 balls with number 20, 4 balls with number 10 and 4 balls with number 5.

A random expreminet consists in drawing simultaneously 3 balls from the urn. Calculate the probability $p$ that the sum of the drawn numbers is equal to 30.
This experiment is repeated 4 times, putting the three balls drawn back into the urn after each trial. Let $X$ be the random variable corresponding to the number of draws giving a sum of 30.
1. What is the distribution of $X$. Give its expected value and standard deviation.
2. Calculate the probability of having at least once the sum 30 out of 4 draws.

Exercice 20 You need someone to help you move. When you call a friend, there’s a one out of four chances he will accept. Let $X$ be the random variable that represents the number of friends you will need to contact for getting help.

What is the probability distribution of $X$.
Calculate $P (X \leq 3)$.
Calculate $E(X)$.

Exercice 21 In order to be selected for the Olympic Games, an athlete must succeed twice in exceeding the minima set by his federation. He has a one in three chance of succeeding in every event he participates in. We consider $X$ the random variable that represents the number of his trials until he is selected.

Determine the probability distribution of $X$.
If this athlete can only participate in four events maximum, what is the probability that he will be selected ?

Exercice 22 A bag contains five tokens: two are numbered 1 and the other three are numbered 2. An unlimited series of draws is made with a token being handed over in the bag. The random variable representing the number of draws before getting a token numbered 1 for the first time is denoted by $Y$.

1. Justify that the random variable $Z = Y + 1$ follows a common distribtion.
2. Deduce the probability distribution of $Y$.
1. Calculate the expected value and variance of $Z$.
2. Calculate the expected value and variance of $Y$.

Exercice 23 The number of power outages that occur in a certain region over a one-year timeframe follows a Poisson distribution of parameter $\lambda=3$.

Calculate the probability that within a period of one year only one failure occurs.
Assuming there is an independence of number of outages from one year to another, calculate the probability that in the upcoming ten years there will be at least one year in which exactly one failure occurs.

Exercice 24 A radio has 2 types of failures: transistor or capacitor. During the first year of use, we note:

$X$ = number of failures due to transistor failure.
$Y$ = number of failures due to capacitor failure.

It is assumed that $X$ and $Y$ are independent random variables following Poisson distribution of respective parameters $\lambda=2$ and $\mu=1$.

Calculate the probability that there are 2 failures due to a transistor failure.
Calculate the probability that there will be at least one failure due to a capacitor failure .
1. What is the distribution of $Z = X + Y$, the number of failures during the first year ?
2. Determine the probability that there will be 2 failures of any type.
3. Calculate $P (Z = 3)$. What do you notice ?
4. Describe the variations of $P (Z = k)$ in function of $k$.
5. Give the average number of failures and the probability that there will be at most one failure during this period.

Solution exercises 23 and 24

Exercices supplémentaires

Exercice 25 Let $X$ and $Y$ two independent random variables such that:

\[P(X=n) = P(Y=n) = \frac{1}{4} (\frac{1+a^n}{n!}) \quad \forall n \in \mathbb{N}\]

Calculate $a$.
Calculate $E(X)$ and $E(Y)$.
Determine the distribution of $Z=X+Y$.

Exercice 26 Let $(X,Z)$ a pair of random variables defined on $\mathbb{N}$. Let

\[p_{k,n} = P(X=k,Z=n)= \frac{\lambda^n e^{-\lambda} \alpha^k (1-\alpha)^{n-k}}{k! (n-k)!} \times {1}_{\{0\leq k\leq n\}}\]

Calculate and recognize the distribution of $Z$. What is the expected value of $Z$.
Calculate and recognize the distribution of $X$.
Calculate $P(X=k | Z=n)$ and recognize its distribution.
Suppose that $Z$ is the number of kids in a famile, $X$ the number of boys while the probability that a kid is a boy is $0.53$. Let $Y$ the number of girls in the family. What is the distribution of $Y$.
Interpret the results.

Exercice 27 A player has a die and a coin. The die is fait while the coin has probability $p$ ($0 < p < 1$) of having a head. We note $q=1-p$. This player throws first the die, then flips the coin as much as the obtained number with the die. At the end, he counts the number of obtained heads. The results of every trial are independant.

Let $X$ the number obtained from the die and $Y$ the number of obtained heads at the end of the game.

Let $(i,j) \, \in \, \{1,\ldots,6\} \times \{0,\ldots,6\}$. Calculate $P(Y=j|X=i)$.
Determine the distribution of $(X,Y)$.
Calculate $P(Y=6)$.
Show that \[P(Y=0)= \frac{q}{6} \big(\frac{1-q^6}{1-q}\big)\]

Remark: $\displaystyle \sum_{i=0}^{n} x^i = \frac{1-x^{n+1}}{1-x}$ if $x\neq 1$.

Given that the player didn’t obtain any head during the game, what is that probability that he obtained 1 by throwing the die ? Calculate this probability if $p=q=\frac{1}{2}$.