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

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

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 Spanish 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 $B \cap C$.

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:

$$(A \cup B) - (A \cap B) = (A \cap \overline{B}) \cup (\overline{A} \cap B)$$

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

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 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.