Events

In the study of everyday phenomena, it is essential to distinguish between certainty and chance. Classify the following experiments as random or deterministic:

1. a) Taking a penalty in a training session and observing whether the ball goes into the goal.
2. b) Measuring the time taken for an object to fall from a height of 5 metres in a vacuum.
3. c) Observing the number of WhatsApp messages you receive between 17:00 and 18:00 on any given Monday.
4. d) Calculating the total cost of buying 3 kilograms of oranges if the price per kilogram is $1,25 \text{ €}$.
5. e) Drawing a card from a Spanish pack of cards and observing whether it is clubs.

We have a wheel of fortune divided into 8 equal sectors, numbered from 1 to 8. The wheel is spun and the number indicated by the arrow is recorded.

Based on this experiment:

1. a) Write down the sample space $E$.
2. b) State the outcomes that form event $A = \text{"getting a number greater than 5"}$.
3. c) State the outcomes that form event $B = \text{"getting an even number"}$.
4. d) Describe the complementary event $\bar{A}$ in words.

A group of friends analyse their hobbies. Let $D$ be the event "playing sport" and $M$ be the event "playing a musical instrument". Express the following situations symbolically using operations on events:

1. a) The friend plays sport or plays an instrument (or both).
2. b) The friend plays sport and also plays an instrument.
3. c) The friend plays sport but does not play any instrument.
4. d) The friend neither plays sport nor plays any instrument.

In a bag there are 10 balls numbered from 1 to 10. We consider the following events:

$A = \{1, 3, 5, 7, 9\}$

$B = \{6, 7, 8, 9, 10\}$

Find the elements of the following resulting events:

1. a) $A \cup B$
2. b) $A \cap B$
3. c) $\bar{A}$ (complementary event of $A$)
4. d) $A - B$ (outcomes that are in $A$ but not in $B$)

Given two events $A$ and $B$ in a random experiment, we know that the probability that neither of them occurs is $P(\bar{A} \cap \bar{B}) = 0,3$. If it is also known that $P(A) = 0,4$ and $P(B) = 0,5$, determine:

1. a) Are $A$ and $B$ incompatible events? Justify your answer.
2. b) Are $A$ and $B$ independent events?
3. c) Calculate $P(A \cap \bar{B})$.

In a survey carried out among 1st-year Bachillerato students about their subscriptions to streaming platforms, it was found that $60\%$ are subscribed to Netflix ($N$), $45\%$ to HBO Max ($H$) and $20\%$ are not subscribed to either of the two. If we choose a student at random, calculate:

1. a) The probability that they are subscribed to both platforms.

1. b) The probability that they are subscribed to Netflix but not to HBO Max ($P(N - H)$).

1. c) Represent the situation using a Venn diagram (draw the sample space $E$ and the sets $N$ and $H$), indicating the probabilities of each region.

Observe the following Venn diagram representing the probabilities of events $A$ (studying French) and $B$ (studying German) in a language centre.

Using the data from the chart, calculate:

1. a) $P(A \cup B)$
2. b) $P(\bar{A} | B)$
3. c) $P(\bar{A} \cap \bar{B})$ using De Morgan's Laws.