Laplace's Rule

A spinner divided into $12$ equal sectors, numbered from $1$ to $12$, is used. Calculate the probability that when the spinner is spun, the pointer stops on:

1. a) A number divisible by $3$.
2. b) A number that is a perfect square (such as $1, 4, 9$).
3. c) A number greater than $2$ and less than $7$.

We toss two fair coins into the air simultaneously.

1. a) Write the sample space of the experiment.
2. b) What is the probability of getting exactly one head?
3. c) What is the probability of getting at least one tail?

In a role-playing game, a player rolls an eight-sided die (octahedron) with faces numbered $1$ to $8$. If the die is fair, calculate the following probabilities using Laplace's rule:

1. a) Getting an even number.
2. b) Getting a number greater than $5$.
3. c) Getting a prime number.

One card is drawn at random from a Spanish deck of $40$ cards. The deck has $4$ suits (oros [coins], copas [cups], espadas [swords] and bastos [clubs]) with $10$ cards each: from $1$ (ace) to $7$, plus the sota (S) [jack], the caballo (C) [knight] and the rey (R) [king]. Determine the probability of the following events:

1. a) The card is an ace.

1. b) It is a cup (copa).

1. c) It is the king of coins (rey de oros).

At a board game festival, two fair six-sided dice (with faces from 1 to 6) are used. A player rolls both dice at the same time. The table shows the sum of the scores for each possible outcome (rows: die 1; columns: die 2).

Calculate the probability that the sum of the scores obtained is a prime number greater than $5$.

In a glass urn for a raffle at a tech store, there are $12$ white balls and $8$ black balls. Three balls are drawn at random, one by one and without replacement. Applying Laplace's rule to the sample space of equiprobable elementary events, determine the probability that the three drawn balls are of the same colour.

In an eSports club, a team of 5 players is to be selected for a national tournament. 15 candidates have entered the selection process: 8 strategy specialists and 7 reflex specialists. The tournament rules require the team to have at least 3 strategy specialists and at least 1 reflex specialist. If the board chooses the 5 members at random from among all the candidates, calculate the probability that the team strictly complies with the rules.

Using the letters of the word $ALGORITMO$, all possible permutations are formed. If one of these arrangements is chosen at random, calculate the probability that the resulting word begins and ends with a consonant, and also that all the vowels appear together somewhere in the string.

An aiming video game consists of throwing a virtual dart at a square target of side length $20\text{ cm}$ that has an inscribed orange circle. The program guarantees that the dart always hits somewhere on the square at random (equally likely). Calculate the probability that the dart hits outside the orange circle. (Consider $\pi \approx 3,14$).

A graphic design application generates a random pattern by placing a point $P$ inside a circle of radius $R = 10 \text{ cm}$. A square of side length $L = 8 \text{ cm}$ is inscribed in the centre of the circle. The application's algorithm considers it a "visual hit" if point $P$ falls inside the square or in the region of the circle above the horizontal diameter but outside the square. Using the concept of geometric probability (extension of Laplace's Rule for areas), calculate the probability of getting a "visual hit" in one attempt.