Exercises: Laplace's Rule

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.

A card is drawn at random from a Spanish deck of $40$ cards. Determine the probability of the following events:

1. a) The card is an ace.
2. b) It is a cup.
3. c) It is the king of coins.

In a drawer there are $10$ black socks, $6$ blue socks and $4$ white socks, all loose. If we pick one at random to go to school, calculate:

1. a) The probability that it is blue.
2. b) The probability that it is not white.
3. c) The probability that it is black or white.

A group of Year 13 students is organising a raffle for the end-of-year trip. They have sold tickets numbered from $1$ to $100$. If the draw is made by picking a number at random, find the probability that the winning number:

1. a) Is a multiple of $10$.
2. b) Ends in the digit $7$.
3. c) Is a number less than $15$.

In a class there are $30$ students, of whom $18$ are girls and $12$ are boys. It is known that $10$ girls and $8$ boys like the math podcast "Derivando". If you organise the data in a contingency table (girls/boys versus like/dislike) and choose a student at random, calculate the probability that the student:

1. a) Is a boy.

1. b) Is a girl who likes the podcast.

1. c) Is a person who does not like the podcast.

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 $7$.

If we choose a letter at random from the word ESTADISTICA, determine the probability of the following events:

1. a) The letter is a vowel.
2. b) The letter is S.
3. c) The letter is a consonant.

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?

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 $5$.

From a Spanish deck of $40$ cards, two cards are drawn simultaneously (which is equivalent to drawing one after another without replacement). Calculate the probability that:

• a) Both cards are of the same suit.

• b) At least one of the cards is an Ace.

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 color.

A survey was conducted at a school on social media use. Of the $100$ Year 13 students, it is known that $65$ use TikTok, $40$ use Instagram and $15$ use neither of the two applications. If we select a student at random, calculate the probability that:

• a) They use both social networks.

• b) They use TikTok but not Instagram.

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 group of $5$ friends (Ana, Borja, Clara, David and Elena) are going to pose in a line for a graduation photo. If they stand at random, calculate the probability that Ana and Elena appear next to each other in the photo.

To log into an online banking app, a random PIN code of $4$ digits (from $0$ to $9$) is generated. Calculate the probability that the assigned code meets the following conditions:

• a) All its digits are distinct.

• b) It is a palindromic number (it reads the same from left to right as from right to left).

In a lottery drum there are $10$ balls numbered from $0$ to $9$. Two balls are drawn successively with replacement (the first is returned before drawing the second). Calculate the probability that the number formed by both digits (considering the first as tens and the second as units) is a multiple of $4$ or a number greater than $75$.

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.

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 center 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.

Three cards are drawn simultaneously from a standard Spanish 40-card deck. Calculate the probability that the sum of the values of the three cards is exactly 10. (For this exercise, consider picture cards: jack, knight, and king, to have values of 8, 9, and 10 respectively, and the ace to be worth 1).

A computer security system requires a 6-character password. The user must choose these characters from the set formed by the 26 lower-case letters of the alphabet and the digits from 0 to 9. If the system automatically generates a password at random, what is the probability that it contains exactly 2 digits, 4 letters and, furthermore, no character is repeated?

We roll three fair six-sided dice (numbered from 1 to 6) and record the outcomes as $(d_1, d_2, d_3)$. Calculate the probability that the product of the three numbers obtained is a perfect square (for example: $1 \times 2 \times 2 = 4 = 2^2$).

At a Year 13 graduation dinner, 8 friends sit at a circular table. Among them are a couple, Alicia and Bruno, who have had an argument and do not want to sit together, and two other friends, Nerea and Thiago, who are inseparable and must sit next to each other. If the seats are assigned completely at random, calculate the probability that the wishes of all four friends are fulfilled simultaneously.

A study is carried out on streaming content consumption habits among young people in a city. It is known that:

• $65\%$ have a subscription to platform N.

• $50\%$ have a subscription to platform D.

• $45\%$ have a subscription to platform H.

• $30\%$ are subscribed to N and D.

• $25\%$ are subscribed to N and H.

• $20\%$ are subscribed to D and H.

• $10\%$ are subscribed to all three platforms.

If a young person is chosen at random, calculate the probability that they are subscribed to only one of the three platforms.

As a hint, only the region for all three platforms has been filled in on the Venn diagram; complete the remaining regions yourself.

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.