Exercises: Random experiments

Classify the following experiments as deterministic or random, briefly justifying your answer in each case:

1. a) Dropping a smartphone from a height of one metre and observing whether it falls towards the ground.
2. b) Opening a playlist in shuffle mode and noting down the first song that plays.
3. c) Calculating the time it takes for a car to travel $200$ km if it travels at a constant speed of $100$ km/h.
4. d) Asking the first person you see in the street in which month they were born.
5. e) Drawing a card from a deck of $40$ cards and observing its suit.

A student has three pens of different colours in their pencil case: Blue (A), Black (N) and Red (R). They decide to draw two pens consecutively without returning the first one to the pencil case.

1. a) Write down the sample space $E$ of this experiment.
2. b) State how many elementary events form this sample space.
3. c) Define the compound event $S = \text{"drawing the blue pen in any position"}$.

We consider the random experiment of rolling a tetrahedral die (4 faces) with faces numbered from $1$ to $4$, and a $1$ euro coin. We record the number on the bottom face of the die and whether the coin shows heads $(C)$ or tails $(X)$.

1. a) Use a tree diagram to represent all possible outcomes.
2. b) Write down the sample space $E$.
3. c) Determine the elements of event $A = \text{"getting an even number on the die and heads on the coin"}$.

In a 2nd Baccalaureate group, subscriptions to streaming platforms are analysed. We let $A$ be the event "being subscribed to Netflix" and $B$ be the event "being subscribed to Spotify". Express in words the meaning of the following events related to set operations:

1. a) $A \cup B$
2. b) $A \cap B$
3. c) $\bar{A}$ (or the complementary event of $A$)
4. d) $A - B$
5. e) $\bar{A} \cap \bar{B}$

A role-playing video game allows you to create a character by choosing between two factions: Warrior (G) or Mage (M). In addition, the character can be from three races: Human (H), Elf (E) or Dwarf (D).

A character is chosen at random from the game server.

1. a) Write down the sample space $E$ using the initial notation (for example, $GH$ for Human Warrior).
2. b) Describe the event $S = \text{"the character is not a Wizard"}$.
3. c) If we let $A$ be the event $\text{"the character is an Elf"}$, state which elements make up the complementary event $\bar{A}$.

Let $E = \{1, 2, 3, 4, 5, 6, 7, 8\}$ be the sample space of a random experiment. Given the events $A = \{1, 3, 5, 7\}$ and $B = \{2, 3, 5, 8\}$, obtain the following sets:

1. a) $A \cup B$
2. b) $A \cap B$
3. c) $A \cap \bar{B}$
4. d) Check whether De Morgan's First Law holds: $\overline{A \cup B} = \bar{A} \cap \bar{B}$.

Observe the following Venn diagram representing the distribution of students playing Basketball ($B$) and Football ($F$) at a municipal sports centre.

From the graph, define the following regions using event operations:

1. a) Students who play both sports.
2. b) Students who play basketball but not football.
3. c) Students who play at least one of the two sports.
4. d) Students who play neither of the two sports mentioned.

We draw a ball from an urn containing $5$ balls numbered from $1$ to $5$. State whether the following events are possible, certain or impossible:

1. a) $A = \text{"getting a number greater than 0"}$.
2. b) $B = \text{"getting the number 7"}$.
3. c) $C = \text{"getting an even number"}$.
4. d) $D = \text{"getting a number less than or equal to 5"}$.
5. e) $F = \text{"getting a negative number"}$.

Hugo, a 6th Year student preparing for the Leaving Certificate, is analyzing the quality control of electronic components in an automotive plant near Limerick. For each component selected at random, he defines the events $A$: "the component has an electrical defect" and $B$: "the component has a mechanical defect". When writing the technical report, Hugo states: "The event consisting of a component having neither of the two defects is formally expressed as $\overline{A} \cup \overline{B}$".\na) Find the conceptual error made by Hugo and explain precisely what type of components are actually represented by the union $\overline{A} \cup \overline{B}$.\nb) Correct his proposal by writing the event "the component has neither of the two defects" in a mathematically rigorous way in two equivalent forms: first as an intersection of events, and then applying De Morgan's Laws from a union.

In a study on public transport punctuality in the Dublin area, the Irish Rail Line 4 commuter service between the stations of Maynooth and Dublin Connolly is analyzed. Consider the following three situations associated with a journey:\n1. Measuring the exact distance in kilometres of the railway route between both stations.\n2. Recording whether a train scheduled at $14.15\text{ h}$ arrives at its destination with a delay of more than $3$ minutes relative to the official schedule.\n3. Calculating the exact time a train will take, knowing in advance that it will travel at a constant speed set automatically by computer at $60\text{ km/h}$ over a straight section of $15\text{ km}$.\na) Classify with reasoning each of the three situations as a deterministic or random experiment.\nb) For situation 2, formally describe its sample space $E_1$. If the train is observed for two consecutive working days noting whether it suffers a delay of more than $3$ minutes ($R$) or is on time ($P$), write the compound sample space $E_2$ and list the elementary events that form the event $S$: "the train is delayed on at most one of the two days".

In a random experiment, the sample space is formed by a set of positive integers $E = \{1, 2, 3, \dots, n\}$. Three events $A$, $B$ and $C$ are defined, about which the following information is known:\n1. $A \cap B = \{4, 6\}$\n2. $A - B = \{2, 8\}$\n3. $B - A = \{1, 3, 5\}$\n4. $C$ is an event mutually exclusive with $A$ ($A \cap C = \emptyset$), but compatible with $B$, with $B \cap C = \{3\}$.\n5. $\overline{A \cup B \cup C} = \{7, 9, 10\}$\n6. The number of elementary events that form $C$ is equal to $3$.\na) Reconstruct with reasoning the sets defining the events $A$, $B$, $C$ and the complete sample space $E$, determining the value of the cardinality $n$.\nb) Express using formal set operations (unions, intersections or complements) and write explicitly the elements of the event: "exactly one of the three events ($A$, $B$ or $C$) occurs".

A team of data analysts at an e-commerce platform based in Ireland classifies each complaint according to two factors: the intake channel (Phone, Web, Mobile App) and the assigned resolution department (Logistics, Billing, Warranties, Customer Service). The random experiment consists of recording both factors for the next complaint received.\na) Without writing the detailed list of cases, make an intuitive estimate of how many elementary outcomes make up the sample space $E$ and how many different events in total (subsets of $E$) can be theoretically defined in this experiment. Check your estimates by calculating the cardinality of $E$ using the multiplication principle and the cardinality of the power set $\mathcal{P}(E)$. Comment on whether the magnitude of the total number of events matches your initial intuition.\nb) The events $A$: "the complaint does not come through the Mobile App" and $B$: "the complaint is assigned to Logistics or Billing" are defined. Describe precisely the verbal meaning of the event $\overline{A \cup B}$, transform it by applying De Morgan's Laws, and determine how many elementary events make it up.\nc) Justify analytically whether the events $A$ and $\overline{B}$ are compatible or mutually exclusive.

In a mobile app development environment, a programmer is testing the security of an access system. A random experiment is defined, consisting of generating a random 4-digit code using only the numbers $\{1, 2, 3, 4, 5, 6\}$. The following events are considered:

• $A$: "The code contains at least two consecutive equal digits".

• $B$: "The sum of the digits of the code is a prime number greater than 15".

• $C$: "The code is a palindrome (it reads the same from left to right as from right to left)".

Describe the sample space $E$ in detail, stating its cardinality, and use the algebra of events to symbolically express the following events:

1. a) The code is not a palindrome or the sum of its digits is not a prime number greater than 15.

1. b) The code is a palindrome and has at least two consecutive equal digits, but the sum of its digits is not a prime number greater than 15.

1. c) Determine whether events $A$ and $C$ are compatible or incompatible, justifying your answer with an example of an element of the sample space.

A communications satellite sends data bursts consisting of binary sequences of length $n=3$. Due to cosmic noise, each bit has a probability of being corrupted. We define the sample space $E$ as the set of all possible binary triples. Let the events be:

• $A = \{(1,0,0), (0,1,0), (0,0,1)\}$

• $B = \{(1,1,0), (1,0,1), (0,1,1), (1,1,1)\}$

Using De Morgan's laws, simplify the expression of event $S = \overline{(\overline{A} \cup B) \cap (A \cup \overline{B})}$ as far as possible. Once simplified, describe in words what common characteristic the sequences forming event $S$ have regarding the number of "ones" present in the burst.

In a genetics laboratory, the mutation of a DNA strand represented by four consecutive nitrogenous bases $\{A, C, G, T\}$ is studied. The experiment consists of observing a sequence of length 3. The following events are considered:

• $M$: "The strand contains at least one $G$ base".

• $N$: "The strand starts with $A$ or ends with $T$".

Determine the number of elements in the following events using counting techniques:

1. $M \cap N$

2. $\overline{M} \cup N$

3. $\overline{M \cup N}$

Explain with reasons whether the set of events $\{M \cap N, \overline{M} \cap N, M \cap \overline{N}, \overline{M \cup N}\}$ forms a partition of the sample space $E$.

Consider a random experiment whose sample space is the set of positive integers, $E = \{1, 2, 3, \dots\}$. We define the events:

$A_k = \{x \in E \mid x \text{ is a multiple of } k\}$ for each $k \in \{2, 3, 5\}$.

Express the following events in terms of $A_2$, $A_3$ and $A_5$, and determine their five smallest elements:

1. a) "The number is a multiple of 2 and of 3, but not of 5".

1. b) "The number is a multiple of only one of the three values $\{2, 3, 5\}$".

1. c) "The number is not a multiple of any of the three values".

A robotic parcel sorting system makes errors randomly. By analysing the history of 1 000 parcels, the following events are defined:

• $A$: "The parcel has an incorrect weight".

• $B$: "The parcel has an illegible barcode".

• $C$: "The parcel has been sent to a wrong route".

It is known that no parcel has all three errors simultaneously ($A \cap B \cap C = \emptyset$). If we denote by $n(X)$ the number of parcels in event $X$, the following hold:

$n(A) = 45$, $n(B) = 38$, $n(C) = 22$, $n(A \cap B) = 12$, $n(A \cap C) = 8$ and $n(B \cap C) = 5$.

Calculate the number of elements in the following events and express them using set operations:

1. $(\overline{A \cup B}) \cap C$

2. $(A \cap \overline{B}) \cup (B \cap \overline{A})$

3. $\overline{A} \cap \overline{B} \cap \overline{C}$

A random experiment is carried out consisting of throwing a dart at a circular target of radius $R$. The result of the experiment is the distance $d$ from the centre of the target to the point of impact. We define the events:

• $S_1$: "The dart hits at a distance $d < \frac{R}{3}$".

• $S_2$: "The dart hits at a distance $\frac{R}{4} < d < \frac{2R}{3}$".

Using interval notation for the value of $d$:

1. Define the sample space $E$.

2. Determine the intervals corresponding to the events $S_1 \cup S_2$, $S_1 \cap S_2$ and $S_1 - S_2$.

3. Find the complementary event of $S_1 \cup S_2$ and explain its physical meaning in the context of the game.

Let $A, B$ and $C$ be any three events of a random experiment. Prove formally, using the properties of the algebra of events (distributive, complementation, etc.), the following identity:

Explain what name this operation is given in set theory and represent the resulting event using a shaded Venn diagram.

In an e-sports tournament, three players ($P_1, P_2, P_3$) face each other in "1 vs 1" knockout matches. The experiment consists of recording the winner of each match until a player wins two consecutive matches or a maximum of 4 matches have been played.

1. Draw the tree diagram representing this random experiment, knowing that in the first game $P_1$ plays against $P_2$, and the loser is replaced by the waiting player ($P_3$).

2. Write down explicitly all the elements of the sample space $E$.

3. Define event $X$: "Player $P_3$ plays at least two games" and event $Y$: "The tournament ends in the third game". Are $X$ and $Y$ mutually exclusive events? Justify your answer by analysing their intersection.

You can do it on paper and send a photo with your answer.

In a new strategic board game, a player must roll a tetrahedral die (faces numbered $1$ to $4$) and, simultaneously, draw a card from a reduced deck containing only the four Aces and the four Kings of a standard pack of cards.

Considering the random experiment of recording the number on the die and the value of the card (Ace or King):

1. a) Describe the sample space $E$ of the experiment.
2. b) Let the events be:

$A$ = "Get an even number on the die".

$B$ = "Get an Ace of any suit".

$C$ = "Get a number less than $3$ on the die and a King card".

Determine the elements comprising the events $A \cup B$, $A \cap B$ and $B \cap C$.

A cybersecurity company analyses access attempts to a server. Consider the sample space $\Omega$ made up of all possible access attempts, which are classified by their state as: $L$ (Legitimate), $S$ (Suspicious) and $M$ (Malicious). It is known that an automatic filtering system classifies access attempts into two categories: $F$ (Filtered) and $P$ (Allowed).

Using Venn diagrams, represent the situation and define the following events symbolically:

1. a) The access attempt is malicious or suspicious, but has been allowed.
2. b) The access attempt is not malicious and has been filtered.
3. c) De Morgan's second law holds for the events "Malicious Access" ($M$) and "Filtered Access" ($F$).

In a robotics laboratory, two independent sensors, $S_1$ and $S_2$, are being tested. The experiment consists of checking whether each sensor triggers ($1$) or not ($0$) when exposed to a stimulus.

1. a) Write down the sample space $E$.
2. b) Describe the complementary event of $A$, where $A$ = "At least one of the sensors triggers".
3. c) If we define event $B$ = "Both sensors act identically", determine $A \cap B$ and the difference event $B - A$.

A teenager decides to organize their playlist for studying. They have $3$ Lo-fi songs, $2$ Jazz songs and $2$ Indie songs. The experiment consists of choosing two songs at random, one after the other, without repetition (without replacement).

1. a) Represent the experiment using a tree diagram, indicating on each branch the possible resulting genres.

1. b) Find the total number of possible outcomes in the sample space (considering the songs as distinguishable elements).

1. c) Describe the compound event "The two chosen songs are of different genres".

We consider the random experiment of rolling an octahedral die (faces from $1$ to $8$) and a standard cubic die (faces from $1$ to $6$). We define the following events:

$A$ = "The sum of the scores is a prime number".

$B$ = "The product of the scores is a multiple of $10$".

1. a) Are $A$ and $B$ compatible or mutually exclusive events? Justify your answer by stating at least one element of the intersection if it exists.
2. b) Calculate the number of elements in the complement of $B$.

A study on the use of social networks is conducted in a sixth-form college. Two students are chosen at random and asked if they regularly use network $X$ and network $Y$. Let $x_i$ be the event "student $i$ uses network $X$" and $y_i$ be the event "student $i$ uses network $Y$".

Express the following events symbolically, using the operations of union, intersection and complementation:

1. a) Only the first student uses both social networks.
2. b) At least one of the two students uses network $X$, but neither uses network $Y$.
3. c) Both students have exactly the same usage habits regarding these two social networks.

A lottery drum contains $10$ balls numbered from $0$ to $9$. Three balls are drawn successively without replacement to form a three-digit number (the first ball corresponds to the hundreds, the second to the tens and the third to the units; outcomes starting with $0$, such as $047$, are allowed).

1. a) How many elements does the sample space $E$ have?
2. b) Describe the event $S$ = "The number formed is a palindrome".
3. c) Describe the event $M$ = "The number formed is a multiple of $5$ and greater than $700$".

Let $E = \{1, 2, 3, 4, 5, 6, 7, 8\}$ be the sample space of a random experiment. We consider the events $A = \{2, 4, 6, 8\}$, $B = \{3, 6\}$ and $C = \{1, 2, 3\}$.

Find the sets resulting from the following operations and check if any inclusion relation holds between them:

1. a) $(A \cup B) \cap C^c$
2. b) $(A \cap C) \cup (B \cap C)$
3. c) $A - (B \cup C)$

Note: $C^c$ represents the complementary event of $C$.

The Central Statistics Office (CSO) designs a study on sustainable mobility habits in which citizens are classified according to three compatible events: $A$ ("regularly uses a bicycle"), $B$ ("regularly uses public transport"), and $C$ ("regularly travels on foot"). For a technical report, an analyst formulates the following user inclusion condition:\n$$S = (A \cap \overline{B}) \cup (A \cap B \cap C) \cup (A \cap B \cap \overline{C})$$\nTo simplify $S$ and deduce which event it is irreducibly equivalent to, two approaches are proposed:\n- Method 1 (Direct algebra of events): Apply distributive, associative, and complement laws ($X \cup \overline{X} = E$).\n- Method 2 (Complementation and De Morgan): Double negate the total expression ($\overline{\overline{S}}$) and transform the unions into intersections of complements.\na) Fully develop Method 1, explicitly indicating which algebraic property justifies each step until obtaining the minimal expression for $S$.\nb) Outline the beginning of Method 2, analyze with reasoning the algebraic difficulties that arise compared to Method 1, and justify which of the two procedures is more efficient and less prone to errors for simplifying this type of expression.\nc) Using the reduced expression obtained in (a), simplify the event $\overline{S \cup B}$ by rigorously applying De Morgan's Laws and interpret its sociological meaning in the context of urban mobility.

EirGrid models the security of the electricity supply between a renewable generation plant $G$ and an industrial estate $P$. The network has three circuit breakers: $I_1$, $I_2$, and $I_3$. Current flows from $G$ to $P$ if at least one path remains closed: a direct path that depends exclusively on breaker $I_1$, and an alternative path formed by breakers $I_2$ and $I_3$ connected in series (both must work simultaneously for the alternative path to conduct power).\nThe failure events of each breaker during an overload are considered: $F_1$ ("breaker $I_1$ fails"), $F_2$ ("breaker $I_2$ fails"), and $F_3$ ("breaker $I_3$ fails").\na) Model in terms of $F_1$, $F_2$, and $F_3$, using event operations ($\cup, \cap, \overline{\phantom{A}}$):\n- The event $C$: "the electricity supply between $G$ and $P$ is completely interrupted".\n- The event $O$: "the electricity supply between $G$ and $P$ remains operational".\nb) Formally prove, using De Morgan's Laws and the distributive properties of the algebra of events, that your expression for $O$ is rigorously equal to the complement of $C$ (that is, $O = \overline{C}$).\nc) Suppose that breakers $I_1$ and $I_2$ share a physical differential system that prevents both from failing simultaneously, so that $F_1$ and $F_2$ are mutually exclusive events ($F_1 \cap F_2 = \emptyset$). Reason whether this technical restriction allows simplifying the Boolean expression of the total outage event $C$, and determine whether the events $F_1$ and $(F_2 \cup F_3)$ are compatible or not under this condition.