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.

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) $F = \text{"getting a negative number"}$.

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 $A = \text{"getting an even number on the die and heads on the coin"}$.

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 $\bar{A}$.

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 $700$".

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.

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) $\bar{A} \cap \bar{B}$

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.

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

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 $C$.

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: $\overline{A \cup B} = \bar{A} \cap \bar{B}$.

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.

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.

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.

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 $B$.

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.