Probability Concepts

Classify the following experiments as random or deterministic, briefly justifying your answer in each case:

1. a) Observing whether the next bus passing the stop in front of the school is route 5.
2. b) Measuring the time it takes for an object to fall to the ground from a height of 2 meters in a vacuum.
3. c) Predicting the winner of the next lottery draw.
4. d) Calculating the cost of 3 kg of apples if we know the price is $90 \text{ \text{RD\$}\ /kg}$.

We are playing with a spinner divided into eight equal sectors, numbered from 1 to 8. Write the sample space $E$ for this experiment and list the elements of the following events:

1. a) $A =$ "Obtaining a number greater than 5".
2. b) $B =$ "Obtaining an even number".
3. c) $C =$ "Obtaining a multiple of 3".

On your cell phone you have 3 social media apps, 2 game apps, and 1 music app installed. When you receive a notification, it is equally likely to come from any of these apps.

1. a) Define the sample space $E$ of this random experiment.
2. b) Describe the event $A$ = "The notification is not from a social media app".
3. c) Write an impossible event related to this context.

We consider the random experiment of choosing a letter at random from the word MATEMATICAS. Define the following concepts associated with this experiment:

1. a) The sample space $E$.
2. b) The event $V =$ "Choosing a vowel".
3. c) The event $C =$ "Choosing a consonant".
4. d) Check whether $V$ and $C$ are complementary events.

In a lottery drum there are balls numbered from 1 to 20. If we draw a ball at random, classify the following events as elementary or compound:

1. a) $S_1 =$ {obtaining the number 13}.
2. b) $S_2 =$ {obtaining a prime number less than 10}.
3. c) $S_3 =$ {obtaining a multiple of 7}.
4. d) $S_4 =$ {obtaining the largest even number in the bag}.

In a sports bag there are 4 red T-shirts, 3 blue ones and 1 white one. If we draw a T-shirt without looking, indicate which of the following options best describes the probability of each event (use the terms: very likely, unlikely, impossible, certain):

1. a) Drawing a green T-shirt.
2. b) Drawing a T-shirt that is not white.
3. c) Drawing a red, blue or white T-shirt.
4. d) Drawing a blue T-shirt.

A die has its faces painted in colors: 3 faces are red, 2 are green and 1 is yellow. If we roll the die:

1. a) What is the probability of "getting red"? Express the result as a fraction.
2. b) What is the probability of "not getting red"?
3. c) What is the technical term for the event in part b) relative to that in part a)?

In a fantasy trading card game, a deck contains the following cards: 8 "Warrior" type, 6 "Wizard" type, 4 "Elf" type and 2 "Dragon" type. If you draw a card at random, calculate the probability that:

1. a) It is a "Wizard".
2. b) It is a "Warrior" or an "Elf".
3. c) It is not a "Dragon".

At a tech fair there is a spinner divided into 12 equal sectors numbered from 1 to 12. The game consists of spinning the spinner and observing the winning number.

1. a) Find the probability of obtaining a prime number.
2. b) Find the probability of obtaining a number greater than 4 and less than 9.
3. c) If we win a prize if a multiple of 3 or an odd number comes up, what is the probability of winning?

One card is drawn at random from a Spanish deck of 40 cards. Given the events $A =$ "Drawing a King" and $B =$ "Drawing a card of Coins", state:

1. a) Are $A$ and $B$ compatible or incompatible events? Why?
2. b) What is the intersection event $A \cap B$?
3. c) Describe in words the union event $A \cup B$.

The figure shows how the deck is organized: 4 suits and 10 cards per suit (numbered from 1 to 7, and the face cards 10 = jack, 11 = knight and 12 = king).

A survey is carried out among the 30 students in a class about their streaming preferences. It is known that 18 students watch Netflix ($N$), 12 watch Disney+ ($D$) and 5 students watch both platforms. If we choose a student at random:

1. a) Use a Venn diagram to represent the situation.

1. b) Calculate the probability that the student watches only Netflix.

1. c) Calculate the probability that the student watches neither of the two platforms (the complement of the union).

We consider the random experiment of rolling two cubic dice and adding their scores. Let the events be:

• $A$ = "The sum is an even number".

• $B$ = "The sum is a multiple of 5".

1. a) Calculate $P(A)$ and $P(B)$.
2. b) Are the events $A$ and $B$ mutually exclusive or not? Justify your answer by finding the elements of $A \cap B$.
3. c) Calculate the probability of the union $P(A \cup B)$.

In a high school in Santo Domingo, $60\%$ of 3rd year of secondary school students use the platform Twitch, $50\%$ use TikTok, and $20\%$ do not use either of them. If we choose a student at random, calculate the probability of the following events:

1. a) That they use both platforms.

1. b) That they use TikTok but not Twitch.

1. c) Are the events "Using Twitch" and "Using TikTok" mutually exclusive? Justify your answer using the probability values.

Two coins are tossed into the air and the result of each one is recorded (C for Heads, X for Tails).

1. a) Use a tree diagram to find all the possible outcomes.
2. b) Write the sample space of the experiment.
3. c) What is the elementary event "getting two heads"?

To go to a festival, Lucía has in her wardrobe 3 sweatshirts (black, white and red) and 2 pairs of trousers (jeans and sweatpants). She chooses a sweatshirt and a pair of trousers at random, with all options equally likely.

1. a) Draw the tree diagram that represents all the possible outfit combinations (sweatshirts and trousers).
2. b) What is the probability that Lucía chooses the red sweatshirt?
3. c) What is the probability that she does not choose the sweatpants?

You can do it on paper and send a photo with your answer.

A group of friends creates a security key for their video game server. The key must have 3 distinct characters chosen from the letters $\{A, B, C\}$ and the numbers $\{1, 2, 3\}$.

1. a) Describe the sample space $\Omega$ if the first character must be a letter and the last character must be a number.

1. b) If a key is chosen at random from all possible keys (without the restrictions of the previous part, but keeping the condition that the 3 characters are distinct), what is the probability that the key consists only of numbers?

Two popular streamers, Alba and Brais, are going to do a collaboration. They have an urn with 4 physical challenge cards and 6 mental challenge cards. They decide to draw two cards successively without replacement.

1. a) Draw a tree diagram representing all the possibilities and assign the probability to each branch.

1. b) Calculate the probability that the second card drawn is a mental challenge, knowing that the first was physical.

1. c) What is the probability that they perform one challenge of each type?

You can do it on paper and send a photo with your answer.

A board game consists of flipping a fair coin (heads and tails are equally likely) and, depending on the result, doing the following:

• If it lands heads, a fair 4-sided die (numbered from 1 to 4) is rolled.

• If it lands tails, a ball is drawn from an urn containing 2 red balls and 3 green balls.

1. a) Represent the experiment using a tree diagram, indicating the probability of each branch.

1. b) Calculate the probability of getting an even number on the die if the coin landed heads.

1. c) What is the total probability that the final result of the experiment is a red ball?

In a tennis tournament final between Carlos and Rafa, the first one to win 3 sets wins. The tournament is interrupted by rain when Carlos is leading 2 sets to 1. Knowing that both have the same probability of winning each remaining set:

1. a) Use a tree diagram to calculate each player's probability of winning the tournament had play continued.

1. b) If the total prize is $12,000$ pesos, how should it be fairly divided according to each player's probability of winning at the moment of the interruption?

At a gaming convention, 200 attendees were classified according to their age and preferred console, obtaining the following contingency table:

If we select an attendee at random, find the probability that they:

1. a) Prefer Nintendo.
2. b) Are over 14 years old.
3. c) Are under 14 years old and prefer PlayStation.
4. d) Are under 14 years old, given that they prefer Nintendo.

In an esports tournament, the peripherals used by 120 participants are analysed. It is known that 75 use a mechanical keyboard and the rest use a membrane keyboard. Of those using a mechanical keyboard, 45 also use a wireless mouse. In total, 65 participants use a wireless mouse.

1. a) Organise these data in a contingency table.

1. b) If we choose a participant at random, what is the probability that they use a membrane keyboard and a wired mouse?

1. c) If we know that a participant uses a wireless mouse, what is the probability that their keyboard is a membrane keyboard?

A content creator has rolled an 8-sided die (octahedron) 500 times to check whether it is loaded. The results obtained for the face with the number "8" are shown in the following table:

1. a) Calculate the relative frequencies for each stage of the experiment.

1. b) What value does the probability of getting an "8" seem to stabilise around?

1. c) If the die were perfect (fair), what should its theoretical probability be according to Laplace's rule?

We roll a six-sided die that we suspect is loaded. After rolling it 500 times, we have obtained the following frequencies: the number 6 came up 125 times, the number 1 came up 35 times, and the numbers 2, 3, 4, and 5 came up with equal frequency among themselves.

1. a) Calculate the relative frequency of the event "Rolling a 6".

1. b) According to the Law of Large Numbers, what probability would you assign to the event "Rolling a number less than 6"?

1. c) If the die were fair, what would be the theoretical probability of rolling a 6? Compare both results.

A delivery drone must land on a square plot of land with side length $10$ meters. Inside the plot there is a circular safety zone (where it must not land) with a radius of $2$ meters, and an optimal landing zone which is a small central square with side length $1$ meter.

Assuming that the drone lands at any point on the plot with equal probability:

1. a) What is the probability that the drone lands in the optimal landing zone?

1. b) What is the probability that it lands outside the circular safety zone? (Use $\pi \approx 3.14$).