Probability Concepts

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

1. a) Observing if the next bus that passes by the stop in front of the school is from line 5.
2. b) Measuring the time it takes for an object to fall to the ground from a height of 2 feet in a vacuum.
3. c) Predicting the winner of the next Christmas lottery draw.
4. d) Calculating the cost of 3 pounds of apples if we know that the price is $1,50 \text{ \$/lb}$.

We are playing with a spinner divided into eight equal sectors, numbered from 1 to 8. Write the sample space $E$ of this experiment and describe 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".

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

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

On your cell phone you have installed 3 social media apps, 2 gaming apps, and 1 music app. When you receive a notification, it can come from any of these apps with the same probability.

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.

In a sports bag, there are 4 red shirts, 3 blue shirts, and 1 white shirt. If we take out a 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) Taking out a green shirt.
2. b) Taking out a shirt that is not white.
3. c) Taking out a red, blue, or white shirt.
4. d) Taking out a blue shirt.

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

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

At a technology 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?

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 if $V$ and $C$ are complementary events.

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 "obtaining the color red"? Express the result as a fraction.
2. b) What is the probability of "not obtaining the color red"?
3. c) What is the technical name for the event in section b) with respect to the one in section a)?

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

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

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.
2. b) Calculate the probability that the student watches only Netflix.
3. c) Calculate the probability that the student watches neither of the two platforms (complementary event to the union).

We consider the random experiment of rolling two six-sided 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 $B$ compatible or incompatible? 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 middle school in Chicago, $60\%$ of 8th-grade students use the platform Twitch, $50\%$ use TikTok and $20\%$ use neither of the two. If we choose a student at random, calculate the probability of the following events:

1. a) They use both platforms.

1. b) 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 flipped and the result is recorded (H for Heads, T for Tails).

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

To go to a festival, Ashley has 3 hoodies (black, white, and red) and 2 pairs of pants (jeans and sweatpants) in her closet.

1. a) Draw the tree diagram representing all the possible clothing combinations (hoodies and pants).
2. b) What is the probability that Ashley chooses the red hoodie?
3. c) What is the probability that she does not choose the sweatpants?

A group of friends creates a security password for their video game server. The password must have 3 different 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 necessarily be a letter and the last one a number.

1. b) If a password is chosen at random from all the possible ones (without the restrictions of the previous section, but keeping the 3 characters different), what is the probability that the password is made up only of numbers?

A content creator has rolled an 8-sided die ($octaedro$) 500 times to check if it is biased. 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.
2. b) Toward what value does the probability of obtaining an "8" seem to stabilize?
3. c) If the die were 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 has appeared 125 times, the number 1 has appeared 40 times, and the numbers 2, 3, 4 and 5 have appeared with the same frequency as each other.

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

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

1. c) If the die were not loaded, what would be the theoretical probability of obtaining a 6? Compare both results.

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

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

1. a) They prefer Nintendo.
2. b) They are over 14 years old.
3. c) They are under 14 years old and prefer PlayStation.
4. d) Knowing that they prefer Nintendo, they are under 14 years old.

In an Esports championship, the peripherals used by 120 participants are analyzed. It is known that 75 use a mechanical keyboard and the rest use a membrane one. Of those who use a mechanical keyboard, 45 also use a wireless mouse. In total, there are 65 participants who use a wireless mouse.

1. a) Organise this 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 membrane?

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

Assuming that the drone lands at any point on the plot with the same 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 $\pi \approx 3,14$).

Two popular streamers, Ashley and Brandon, 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 and without replacement.

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

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

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

A board game consists of flipping a coin and, depending on the result, doing the following:

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

• If it lands on tails, a marble is drawn from a bag containing 2 red marbles and 3 green marbles.

1. a) Represent the experiment using a tree diagram.

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

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

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

1. a) Use a tree diagram to calculate the probability each one has of winning the tournament if they had continued playing.

1. b) If the total prize is $\$12.000$, how should it be shared fairly according to the probabilities of victory of each one at the time of the interruption?