Theoretical Probability

A spinner divided into $12$ equal sectors, numbered from $1$ to $12$, is used. Calculate the probability that when spinning the spinner, the arrow 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 simultaneously.

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

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

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

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.

In a drawer there are $10$ black socks, $6$ blue socks and $4$ white socks, all of them loose. If we choose one at random to go to class, 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 high school seniors organize a raffle for their end-of-year trip. They have sold tickets numbered from $1$ to $100$. If the draw is made by extracting 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 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 a class there are $30$ students, of which $18$ are girls and $12$ are boys. It is known that $10$ girls and $8$ boys like the math podcast "Derivando". If a student is chosen at random, calculate the probability that:

1. a) They are a boy.
2. b) They are a girl who likes the podcast.
3. c) They are a person who does not like the podcast.

A survey has been carried out in a high school on the use of social networks. Of the $100$ seniors, 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.

A study is carried out on the streaming content consumption habits of 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.

At a board game festival, two fair cubical dice (with faces from 1 to 6) are used. A player rolls both dice at the same time. Calculate the probability that the sum of the scores obtained is a prime number greater than $5$.

Two cards are drawn simultaneously from a Spanish deck of $40$ cards (which is equivalent to drawing one after the other 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 an electronics store prize draw, 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 balls drawn are of the same color.

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

A group of $5$ friends (Ashley, Brandon, Chloe, David, and Emma) are going to pose in a row for a graduation photo. If they are positioned at random, calculate the probability that Ashley and Emma appear together in the photo.

At a senior graduation dinner, 8 friends sit at a circular table. Among them are a couple, Ashley and Brandon, who have argued and do not want to sit together, and two other friends, Chloe and David, 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.

Using the letters of the word $ALGORITMO$ (all of them distinct), 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 vowel, and that all the vowels also appear together somewhere in the string.

To log into an online banking application, a random $4$-digit PIN code (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).

A computer security system requires a 6-character password. The user must choose these characters from the set formed by the 26 lowercase 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 that, furthermore, no character is repeated?

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

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

A target-shooting video game consists of throwing a virtual dart at a square target of side $20\text{ cm}$ which has an inscribed orange circle. The program guarantees that the dart always hits a point on the square randomly (equiprobably). Calculate the probability that the dart hits outside the orange circle. (Consider $\pi \approx 3,14$).

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