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.

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).

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 maths 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?

Hugo wants to calculate the probability that, when tossing three fair $1$-euro coins simultaneously, exactly two heads and one tail are obtained. Hugo reasons as follows: «When tossing three coins, the total number of heads obtained can only be $0$, $1$, $2$, or $3$. Therefore, the sample space has $4$ possible outcomes. Since the event "obtaining two heads" is one of those four outcomes, applying Laplace's Rule the probability is $P = \frac{1}{4} = 0,25$».

1. a) Identify and explain in detail the conceptual error made by Hugo, referring explicitly to the indispensable condition that Laplace's Rule requires regarding elementary events.
2. b) Write down the correct sample space formed by equiprobable events and calculate the actual value of the probability of obtaining exactly two heads.
3. c) Calculate the probability of obtaining at least two heads.

An opaque urn used in a raffle to allocate spots for extracurricular activities contains only blue slips and red slips. In total, there are $50$ slips identical to the touch. It is known that, if a slip is drawn at random, the probability of it being blue is exactly $0,36$.

1. a) Determine how many slips of each color are initially in the urn, justifying the calculation using Laplace's Rule.
2. b) Set up an equation and calculate how many additional blue slips must be placed into the urn, without removing any of the existing ones, so that the probability that a randomly drawn slip is blue rises to exactly $0,60$.

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 colour.

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 a university Faculty of Social Sciences, $10$ student applications have been received for an Erasmus exchange stay: $7$ are studying for a Degree in Business Administration and Management (ADE) and $3$ are studying for a Degree in International Relations. Since the academic records are identical, the dean's office decides to award the $4$ available spots by raffle, selecting $4$ of the $10$ applicants at random and simultaneously.

1. a) Before performing any analytical calculation, estimate whether the probability that all $4$ spots go exclusively to ADE students is less than $0,15$, between $0,15$ and $0,35$, or greater than $0,35$. Briefly justify your intuitive prediction taking into account that $70\,\%$ of the applicants belong to ADE.
2. b) Apply Laplace's Rule using combinations to calculate the exact value of this probability.
3. c) Compare your prior estimate with the exact result obtained. From this calculation, deduce the probability that the selected delegation includes at least one International Relations student.

According to the railway company's records, a Cercanías commuter train from Aranjuez arrives at Atocha station in Madrid at a random instant uniformly distributed between 8:00 and 8:20 in the morning. A passenger needs to transfer to another train bound for Nuevos Ministerios that departs from Atocha at an instant, also random and uniform, between 8:15 and 8:35. To walk between platforms and access the new train before the doors close, the passenger strictly requires a minimum time margin of $10$ minutes between the arrival of her first train and the departure of the second.

Model the situation using the continuous version of Laplace's Rule (geometric probability):

1. a) Define the variables $x$ and $y$ (minutes elapsed since 8:00 for the arrival of the first train and the departure of the second, respectively), algebraically describe the two-dimensional sample space, and calculate its total area.
2. b) Write the inequality that describes the time condition that must be met for the passenger to make the transfer successfully, and determine the area of the favorable region in the plane.
3. c) Calculate the probability that the passenger manages to make the transfer on time and interpret the result.

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 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.

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: $1 \cdot 2 \cdot 2 = 4 = 2^2$).

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.

In a competitive examination process to enter the body of inspectors of the Public Administration in Spain, a panel examines the candidates orally by randomly drawing $3$ distinct topics from an official syllabus of $25$ topics. A candidate has thoroughly prepared $20$ topics and has not been able to study the remaining $5$. We want to calculate the probability that the candidate will have to face at least one topic she has not prepared.

During a study session, three opinions arise among the candidates:

• Daniel claims: "Since the topics are drawn one by one from a bag, it is mandatory to take the drawing order into account and use permutations without repetition (variaciones), both for favorable and possible outcomes. Working with combinations is incorrect because it ignores the physical order of the draw".

• Lucía proposes calculating the probability using combinations by summing the cases of three mutually exclusive situations: getting exactly one unprepared topic, two unprepared, or all three unprepared.

• Mateo proposes first calculating the probability of the complementary event ("all three drawn topics are from those the candidate has studied") using combinations and subtracting the resulting value from one.

1. a) Critically analyze Daniel's objection: is it necessary to take order into account to apply Laplace's Rule in successive draws without replacement? Justify algebraically why the approach with combinations and the approach with permutations lead to exactly the same result, relying on the principle of consistency of criterion.
2. b) Compare Lucía's and Mateo's strategies. Reason which of the two methods is preferable regarding calculation efficiency and the probability of making mistakes.
3. c) Apply the most efficient method to determine the exact probability that the candidate will have to present at least one unprepared topic.

The general assembly of an olive oil cooperative in Jaén must renew its delegate committee by randomly choosing $3$ members from a closed group of $10$ candidates. Among the candidates, there are $k$ traditional olive grove producers and $10 - k$ intensive olive grove producers (with $k \ge 3$). The election is conducted by simultaneously drawing $3$ ballots from an urn.

It is known that, after applying Laplace's Rule, the probability that the delegate committee has at least one representative from intensive olive farming is exactly equal to $\frac{5}{6}$.

1. a) Set up a combinatorial equation that allows finding the value of $k$, using the analysis of the complementary event and combinations.
2. b) Solve the previous equation with reasoning and determine how many candidates from each farming modality ran for election.
3. c) Suppose that, before the vote, the regulations are modified so that the delegate committee now has $5$ members instead of $3$ (keeping the same group of $10$ candidates with the found value of $k$). Calculate the probability that intensive olive grove producers achieve an absolute majority on the new committee (that is, at least $3$ of the $5$ seats).