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 winning number of the next Lotería Nacional draw.
4. d) Calculating the cost of 3 kg of apples if we know the price is $\$30$ per 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".

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}.

A card is drawn at random from a baraja española, a 40-card deck (four suits: coins, cups, swords and clubs; each suit has the cards 1 to 7, a jack, a knight and a king). Given the events $A =$ "Drawing a king" and $B =$ "Drawing a card of the coins suit", indicate:

1. a) Are $A$ and $B$ mutually exclusive events or not? 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.

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).

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)?

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 $P(A \cup B)$.

Two coins are tossed in the air and the outcome is recorded (H for heads/águila, T for tails/sol).

1. a) Use a tree diagram to find all 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.