Exercises: Events

Classify the following experiments as deterministic or random:

1. a) Measuring the time it takes for an object to fall from $10$ metres in a vacuum.
2. b) Recording the colour of the next car that passes along the street.
3. c) Calculating the interest generated by a fixed-term bank deposit knowing the principal and the interest rate.
4. d) Drawing a ball from a lottery drum.

A counter for each letter of the word "ALGORITMO" is placed in a bag. If the experiment consists of drawing a counter at random and recording the letter obtained, write down the sample space $E$.

In the experiment of spinning the following spinner and recording the number on which it lands, we consider events $A = \{1, 2\}$ and $B = \{2, 3, 4\}$.

Write down the elements that form the union event $A \cup B$ and the intersection event $A \cap B$.

When drawing a card from a Spanish deck of $40$ cards, we consider event $C = \{\text{drawing a face card}\}$. Describe in words what the complementary event $\bar{C}$ means and how many elements of the sample space make it up.

A die with $8$ faces numbered from $1$ to $8$ is rolled. State whether the following pairs of events are compatible or incompatible:

1. a) $A = \{\text{drawing an even number}\}$ and $B = \{\text{drawing a multiple of } 3\}$
2. b) $C = \{\text{drawing a number greater than } 6\}$ and $D = \{\text{drawing a number less than } 5\}$

In an urn there are $5$ red balls and $3$ green balls. If we draw a ball at random and observe its colour, invent a certain event and an impossible event.

Consider the experiment of choosing an integer at random from $1$ to $10$. Let $A = \{2, 4, 6, 8, 10\}$ and $B = \{1, 2, 3, 4, 5\}$ be the events. Determine the elements of the difference event $A - B$.

Write down the sample space corresponding to the random experiment consisting of flipping a one-euro coin and a two-euro coin simultaneously, recording whether heads ($C$) or tails ($X$) is obtained on each.

In the Madrid Cercanías commuter rail network, three situations related to line C-5 are analyzed at Atocha station:

1) The time it takes for a train to brake from $60\text{ km/h}$ to a complete stop if the driver applies the service brake with a constant deceleration of $1,2\text{ m/s}^2$ on a straight, dry track.

2) The number of minutes of delay with which the next train scheduled for $08:15$ arrives at the platform.

3) The type of transit pass/ticket that the next passenger validates at the turnstile: Youth Pass ($J$), Single Ticket ($S$), or 10-trip Multi Card ($M$).

1. a) Classify each of the three situations as a deterministic or random experiment, briefly justifying your decision in each case.
2. b) For situation 3, write the sample space $E$.
3. c) In situation 3, define the event $A = \text{“the passenger does not use a single ticket”}$ by listing its elements. Then, write the elements of the complementary event $\bar{A}$ and indicate whether $\bar{A}$ is an elementary or compound event.

Pablo is solving an assignment about rolling a fair six-sided die with faces numbered from $1$ to $6$, whose sample space is $E = \{1, 2, 3, 4, 5, 6\}$. Two events are defined: $A = \text{“getting a number less than or equal to } 2\text{”}$ ($A = \{1, 2\}$) and $B = \text{“getting a multiple of } 5\text{”}$ ($B = \{5\}$).

Pablo writes the following conclusion in his notebook:

“Since events $A$ and $B$ have no elements in common, that is, $A \cap B = \emptyset$, it necessarily follows that $B$ is the complementary event of $A$ ($B = \bar{A}$), so they are mutually exclusive and at the same time complementary events.”

1. a) Find and explain the conceptual error that Pablo makes when deducing that $B$ is the complement of $A$. What fundamental condition is missing for the pair of events to be complementary?
2. b) Correct Pablo's answer: write what the complementary event $\bar{A}$ actually is by listing its elements.
3. c) Write an example of an event $C$ (different from $\emptyset$) that is mutually exclusive with $A$, but is not its complementary event either.

In a quality control check at an electronic components factory, parts are drawn one by one until the first defective part is found, at which point the process stops. By regulations, a maximum of $3$ draws is performed. If $D$ represents a defective part and $N$ a non-defective part, write down the sample space of this experiment.

In a market study on vehicle preferences, events $A = \{\text{el cliente prefiere coche eléctrico}\}$ and $B = \{\text{el cliente prefiere coche automático}\}$ are defined. Express in words the meaning of the following events:

1. a) $A \cap \bar{B}$
2. b) $\overline{A \cup B}$

The following Venn diagram represents a sample space $E$ and two events $A$ and $B$. The numbers $1, 2, 3, 4$ indicate the different regions of the diagram.

Identify which region or regions (indicated by their numbers) correspond to the following events:

1. a) $A \cap B$
2. b) $\bar{A} \cap B$
3. c) $\overline{A \cup B}$

Two dice shaped like regular tetrahedra, whose faces are numbered from $1$ to $4$, are rolled. The experiment consists of recording the sum of the scores obtained on the hidden faces (the ones touching the table).

1. a) Write down the sample space $E$.
2. b) List the elements of event $S = \{\text{the sum is a prime number}\}$.

At a weather station, events $L = \{\text{it is raining today}\}$ and $V = \{\text{it is windy today}\}$ are defined. Write the following situations using mathematical operations with events (union, intersection, complement):

1. a) Today it is raining but it is not windy.
2. b) Today it is neither raining nor windy.
3. c) Either it is raining or it is windy, but not both at the same time.

Two cards are drawn simultaneously from a Spanish deck of $40$ cards. We consider events $O = \{\text{at least one card is a gold}\}$ and $R = \{\text{at least one card is a king}\}$.

Are $O$ and $R$ compatible events? Justify your answer by describing an elementary event that belongs to the intersection $O \cap R$.

A logistics company classifies received parcels according to their weight ($L$: light, $P$: heavy) and according to their destination ($N$: national, $I$: international). A parcel is chosen at random. Determine whether events $A = \{\text{the parcel is light and national}\}$ and $B = \{\text{the parcel is heavy and international}\}$ are compatible or incompatible, justifying your answer based on the sample space.

A traditional cubic die is rolled. We consider events $A = \{1, 2, 3\}$, $B = \{2, 4, 6\}$ and $C = \{4, 5, 6\}$.

Find the elements that form the following events:

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

From a representative sample of $100$ young people aged $18$ to $24$ in Spain, based on data from the Labour Force Survey (EPA) of the INE, their education and employment status is analyzed. The events $E = \text{“the person is enrolled in formal education”}$ and $T = \text{“the person is employed”}$ are defined. The data shows that $62$ young people study, $36$ work, and $14$ combine studying and working at the same time.

1. a) Before performing any exact calculation, make a reasoned estimate: do you think the number of young people who neither study nor work (event $\bar{E} \cap \bar{T}$) will be greater than $25$ or less than $25$? Explain what you base your intuitive estimate on.
2. b) Describe in your own words what the events $E \cup T$ and $E - T$ represent in the context of this sample.
3. c) Calculate the exact number of young people belonging to $E \cup T$ and to $\bar{E} \cap \bar{T}$. Check whether your estimate from section a) was correct and reflect on the result obtained.
4. d) Justify using the mathematical definition whether events $E$ and $T$ are compatible or mutually exclusive.

In a charity raffle organized at a school, a ball is drawn from an urn containing $10$ balls numbered from $1$ to $10$, so the sample space is $E = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.

We do not directly know events $A$ and $B$, but we know that they simultaneously satisfy the following four conditions:

1. $A \cap B = \{3, 7\}$

2. $A - B = \{1, 5, 9\}$

3. $A \cup B = \{1, 2, 3, 5, 7, 8, 9\}$

4. All elements of event $B$ are odd numbers except two.

1. a) Reconstruct events $A$ and $B$ with reasoned arguments, detailing which elements each one contains.
2. b) Determine the elements of the complement of the union, $\overline{A \cup B}$, and interpret what this event means in the context of the raffle.
3. c) Check whether the events $(A - B)$ and $B$ are compatible or mutually exclusive. Justify your answer by reasoning about the definition of both operations.

Prove, using the logic of set operations, that for any two events $A$ and $B$, event $(A \cup B) \cap (A \cup \bar{B})$ is equivalent to event $A$.

A board game consists of flipping a coin repeatedly until two consecutive heads or two consecutive tails are obtained. Write down the sample space of this random experiment. Is it a finite or infinite sample space?

The following Venn diagram shows three events $A$, $B$ and $C$ within a sample space $E$. The region shaded in orange represents a specific operation between these events.

Write the mathematical expression that defines the shaded region using the operations of union, intersection and/or complement. How would this event be expressed in words?

In a plant genetics study, three characteristics of a plant are analysed: pest resistance ($R$), drought tolerance ($T$) and high yield ($P$). Express the following situations using operations with events:

1. a) The plant has exactly two of the three characteristics.
2. b) The plant has at least one of the three characteristics, but is not resistant to pests.

Let $A$ and $B$ be two events of a random experiment. Simplify the following event expression using the properties of operations (commutative, associative, distributive, De Morgan's laws, etc.):

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

In an air traffic control system, three radars $R_1$, $R_2$ and $R_3$ detect the presence of aircraft. We know by design that it is impossible for all three radars to fail simultaneously, but any pair of radars can fail at the same time. If event $F_i$ represents "radar $i$ fails", translate this information into mathematical expressions using unions and intersections of the events $F_1, F_2, F_3$.

In the experiment of drawing a card from a French deck of $52$ cards, we define the events:

$C = \{\text{the card is a heart}\}$

$D = \{\text{the card is a diamond}\}$

$F = \{\text{the card is a face card (J, Q, K)}\}$

Determine the elements that make up event $(C \cup D) - F$ and describe it in words.

Consider a random experiment with a sample space $E$. If $A, B$ and $C$ are three events such that they form a partition of the sample space (that is, they are pairwise mutually exclusive and their union is the certain event $E$), determine which event the following simplified expression is equivalent to:

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

In a local padel tournament, a pair plays two consecutive matches. In each match, there are two possible outcomes: winning ($G$) or losing ($P$). If the pair wins, it is recorded whether it was in straight sets ($G_2$) or requiring the tiebreaker set ($G_3$); if they lose, it is recorded whether they won at least one set ($P_1$) or none ($P_0$).

We wish to study the complete sample space $E$ and, in particular, the event $S = \text{“the pair suffers no straight-set defeat (that is, in neither match do they lose by } P_0\text{)”}$.

Two classmates discuss the best way to organize the information:

• Clara proposes using a tree diagram, branching first on the result of the first match (with $4$ branches) and, from each of them, branching another $4$ branches for the second match.

• Daniel proposes constructing a $4 \times 4$ two-way table, placing the four possible outcomes of the first match in the rows and those of the second in the columns.

1. a) Represent the sample space $E$ using Daniel's method (the two-way table) and state how many elementary events make up $E$.
2. b) Mark on the table and write the elements that form event $S$.
3. c) Compare both methods: analyze the advantages and disadvantages of the tree diagram versus the two-way table for this two-stage experiment with $4$ options each. Justify which of the two methods you consider clearer and less prone to error for identifying events such as $S$.
4. d) Write in words what the complementary event $\bar{S}$ means and determine its elements. Is it faster to define $S$ by identifying its cases directly or by first identifying those of $\bar{S}$? Reason your answer.

In the Low Emission Zones (ZBE) regulated by the DGT in Spanish cities such as Madrid or Sevilla, vehicles entering the city center are monitored. Each vehicle is classified according to two variables: its environmental badge, which belongs to $D = \{0, E, C, B, A\}$ (where $0$ is Zero emissions, $E$ is Eco, $C$ and $B$ are combustion according to Euro standards, and $A$ represents vehicles without a badge), and the parity of the last digit of its license plate, $M = \{P, I\}$ (Even or Odd). Each elementary event of the experiment is a pair $(d, m)$.

The following events are defined:

• $V_1$: “The vehicle has a Zero or Eco sticker” ($d \in \{0, E\}$).

• $V_2$: “The vehicle has a $B$ or $C$ sticker”.

• $V_3$: “The vehicle has an even license plate” ($m = P$).

• $R$: “The vehicle is restricted from entering during a high-pollution episode”, a protocol that affects all vehicles without a sticker ($A$), and also vehicles with a $B$ sticker whose license plate is odd.

1. a) Write the sample space $E$ detailing its elementary events. How many elements does it have?
2. b) Express event $R$ formally using set operations (union, intersection, difference, or complement) from events $V_1$, $V_2$, $V_3$ or from elementary events.
3. c) Mathematically analyze whether events $V_1$ and $R$ are mutually exclusive. Are they also complementary events ($V_1 = \bar{R}$)? Rigorously justify your answer based on the elements of the sample space.
4. d) A new municipal ordinance proposes: “All vehicles belonging to $(V_2 \cap V_3) \cup V_1$ may drive”. Translate this rule with complete precision into the language that a driver would understand on a DGT informational sign.