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 metres in a vacuum.
3. c) Predicting the winner of the next Christmas Lottery draw.
4. d) Calculating the cost of 3 kg of apples if we know the price is $1,50 \text{ €/kg}$.

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}.

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).

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.

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 $P(A \cup B)$.