Exercises: Counting techniques

A streaming platform allows users to personalize their profile by choosing an avatar from 4 different animals (lion, panda, cat and wolf) and a background color from 3 options (blue, green and orange).

1. a) Describe the sample space $E$ of the possible profile combinations.
2. b) How many elements does the sample space have? Use the multiplication principle to justify this.

On the menu at your school cafeteria you can choose a starter from salad, pasta or rice; a main course from chicken or fish; and a dessert from fruit or yogurt.

1. a) Represent all the possible full menu options using a tree diagram.
2. b) If you choose a menu at random, what is the probability that it includes "pasta" as a starter?

A combination padlock for a locker has a 3-digit code. Each digit can be a number from $1$ to $5$.

1. a) How many different combinations can be created if the numbers can be repeated?
2. b) How many combinations are there if the three numbers must be different?

In an 11th-grade class there are 30 students. It is known that 18 of them have passed the last math exam and 12 have passed the English exam. If there are 5 students who have passed both subjects:

1. a) Calculate the probability that, when choosing a student at random, they have passed math or English.
2. b) Calculate the probability that they have not passed either of the two subjects.

Five friends (Ashley, Brandon, Clara, David, and Elena) go to the movies and sit in a row of five consecutive seats.

1. a) In how many different ways can they sit in the row?
2. b) If they decide that Ashley and Brandon must always sit at the two ends of the row, how many different arrangements are possible now?

A ball is drawn from an urn containing 10 balls numbered from $1$ to $10$. We consider the following events:

$A = \text{"Getting a prime number"}$

$B = \text{"Getting a multiple of 3"}$

1. a) Write down the elements that make up events $A \cup B$ and $A \cap B$.
2. b) Calculate the probabilities $P(A)$, $P(B)$ and $P(A \cup B)$ applying Laplace's rule.

A music band has prepared 10 songs for their new album, but the record label tells them they can only include 4 of them.

1. a) How many different selections of 4 songs can they make if the order does not matter?
2. b) What are these types of groupings called in combinatorics? Justify your answer.

8 players participate in a local chess tournament. Three different trophies will be awarded for first, second, and third place.

1. a) In how many different ways can the three prizes be awarded?
2. b) If last year's champion participates again and we are certain they will finish in the top three, how many podium classifications are possible now?

A group of 12 11th-grade students has been selected to attend a sustainability conference. The school must choose 4 of them to form a committee of spokespersons.

• a) In how many different ways can the committee be chosen if there is no distinction of roles?

• b) If among the 12 students there are 7 girls and 5 boys, what is the probability that the committee is formed by exactly 2 boys and 2 girls?

A streaming content platform uses a security code for parental control. The code consists of 2 letters (from a 26-letter English alphabet) followed by 3 digits (from 0 to 9).

• a) How many different codes can be created if letters and numbers can be repeated?

• b) If a user forgets their code but remembers that it starts with the letter "A" and ends with the number "7", how many possible combinations would they have to try?

• c) If the code were generated randomly without allowing repetition of any character (letters or numbers), what would be the probability of the code being "BX123"?

In a local basketball league, the effectiveness of a player's three-point shots has been analyzed. It is known that if they score the first three-pointer, the probability of scoring the second is $0,7$. However, if they miss the first, the probability of scoring the second is only $0,4$. It is known that the probability of scoring the first three-pointer of a game is $0,5$.

• a) Draw the tree diagram associated with the experiment of shooting two consecutive three-pointers.

• b) Calculate the probability that the player scores exactly one of the two shots.

• c) If we know that the player has missed the second shot, what is the probability that they had scored the first?

A technology company manufactures wireless earphones in two plants, one in the US and another in Vietnam. The plant in the US produces $40\%$ of the total and Vietnam the rest. It has been detected that $3\%$ of the earphones from the US have connection faults, while in the Vietnam plant this fault appears in $6\%$ of the production.

• a) If we buy a pair of earphones at random, what is the probability that they work correctly?

• b) If we buy a pair of earphones and they turn out to have connection faults, what is the probability that they were manufactured in the plant in the US?

A survey has been carried out among 200 young people about their reading habits and use of social media. The data is collected in the following contingency table:

If we choose a young person at random from this survey, calculate:

• a) The probability that they read regularly and use TikTok for less than 2 hours a day.

• b) Given that the young person reads regularly, what is the probability that they use TikTok for more than 2 hours a day?

• c) Are the events "Reading regularly" and "Using TikTok for more than 2 hours a day" independent? Justify the answer mathematically.

In a toolbox there are 5 Phillips screwdrivers and 3 flathead screwdrivers. 2 screwdrivers are chosen at random, one after the other and without returning the first one to the box.

• a) Calculate the probability that both are of the same type.

• b) Calculate the probability that the second is a flathead given that the first was a Phillips.

• c) If the experiment were done with replacement, how would the probability in part a) change?

Let $A$ and $B$ be two events in a sample space such that we know their probabilities: $P(A) = 0,6$, $P(B) = 0,4$ and $P(A \cap B) = 0,2$.

Calculate the following probabilities using the properties of probability:

• a) $P(A \cup B)$

• b) $P(\bar{A} \cap \bar{B})$ (Probability that neither A nor B occurs).

• c) $P(A | \bar{B})$ (Probability of A given that B does not occur).

A fashion designer is creating a capsule collection. For the final fashion show, they have to organize 6 models in a single line.

• a) In how many different ways can the 6 models walk the runway?

• b) If among the models there are two who are sisters and the designer insists that they must walk one immediately after the other, in how many ways can the line be organized now?

• c) If the designer decides that the two sisters cannot be together under any circumstances, how many options do they have?

A computer security system requires passwords to have exactly $8$ characters. These must be chosen from the $26$ letters of the alphabet (case-insensitive) and the $10$ numerical digits ($0-9$). A password is considered "strong" if it contains at least two numerical digits and does not repeat any character. If a program generates a random password that meets the condition of not repeating characters, calculate the probability that it is "strong".

In an inter-school debate tournament, $12$ students participate: $7$ girls and $5$ boys. For the final stage, a team of $4$ representatives must be chosen at random. Calculate the probability that the chosen team simultaneously satisfies the following:

• There is representation of both sexes.

• The number of girls is strictly greater than the number of boys.

A logistics company organizes its routes using a barcode of $10$ vertical stripes. Each stripe can be black, gray or white. For scanner readability reasons, codes with three consecutive stripes of the same color are not allowed.

1. How many different codes of $10$ stripes can be formed in total (without restrictions)?

2. If a code is chosen at random from all possible ones (without restrictions), what is the probability that the stripes in the odd positions are all black and those in the even positions are all white or gray?

A data analyst studies user behavior on a streaming platform. They have the following information about a sample of $500$ users:

• $65\%$ are under $25$ years old.

• $40\%$ of the users use the platform mainly to watch documentaries.

• Among those who do not watch documentaries, $70\%$ are under $25$ years old.

If a user is selected at random and turns out to be under $25$ years old, calculate the probability that their main preference is documentaries. Use a contingency table to organize the information.

In a toolbox there are $15$ LED bulbs, of which it is known that $3$ are defective. If we extract $4$ bulbs one by one and without replacement:

1. Calculate the probability that the fourth bulb extracted is the second defective one to appear in the process.

2. If the extractions were done with replacement, would the previous probability change? Justify the answer using counting techniques.

It is known that in a certain city it rains on $20\%$ of the days ($LL$), it is very windy on $10\%$ ($V$) and both conditions occur simultaneously on $5\%$.

1. Calculate the probability that on a randomly chosen day it does not rain and it is not windy.

2. Given that it has not rained on a certain day, what is the probability that it was very windy?

3. Are the events "raining" and "being very windy" independent? Prove it mathematically.

A multiple-choice exam consists of $5$ questions with $4$ answer options each, of which only one is correct. A student who has not studied at all decides to answer at random. However, they have a strategy: for each question, they discard one of the options that they know for sure is false (because it is absurd) and choose at random from the remaining three.

1. Calculate the probability that they get exactly $3$ correct answers.

2. What is the probability that they get at least one question right?

An urn contains $n$ white balls and $6$ red balls. Two balls are extracted simultaneously (or successively without replacement). If it is known that the probability of both balls being white is equal to $1/3$, determine the total number of balls in the urn. (Note: you will need to set up a combinatorial equation and solve it to find the value of $n$).