Exercises: Events

A 2nd-year Bachillerato student is organizing their digital folders of notes. In their "Economía" folder they have 12 PDF files, 8 text documents and 5 diagrams in image format. If they pick a file at random to review before an exam, determine:

1. a) The sample space of the experiment.

1. b) The probability that the chosen file is a diagram in image format.

1. c) The probability that the chosen file is not a text document.

In a Year 13 class, a survey on social media use was conducted. Let event $A$ be "using Instagram" and event $B$ be "using TikTok". Define the following situations using set operations on events (union, intersection or complement):

1. a) The student uses both social networks.
2. b) The student uses at least one of the two networks.
3. c) The student does not use Instagram.
4. d) The student uses Instagram but not TikTok.

Consider two events $A$ and $B$ associated with a random experiment. If we know that $P(A) = 0.6$, $P(B) = 0.4$ and $P(A \cap B) = 0.2$, calculate the following probabilities:

1. a) $P(A \cup B)$

1. b) $P(\bar{A})$ (probability of the complement of event $A$).

1. c) $P(\bar{A} \cap \bar{B})$ using De Morgan's Laws.

The Venn diagram shows only the data given in the statement.

An esports team has players of three nationalities on its roster: 5 Mexican, 3 French, and 2 Italian. If a captain is randomly selected for the next tournament, calculate:

1. a) The probability that they are French.
2. b) The probability that they are not Mexican.
3. c) Are the events "being Mexican" and "being Italian" compatible or incompatible? Justify your answer.

Given two events $M$ and $N$ that are incompatible with each other, where $P(M) = 0.35$ and $P(N) = 0.45$, find the probability that:

1. a) Event $M$ or event $N$ occurs.
2. b) Neither of the two events occurs.
3. c) $M$ and $N$ occur simultaneously.

On a streaming platform, $70\%$ of users watch action series ($S$), $40\%$ watch documentaries ($D$) and $25\%$ watch both types of content. Calculate the probability that a randomly chosen user:

1. a) Watches action series or documentaries.

1. b) Watches neither of these two types of content.

Let $A$ and $B$ be two events such that $P(A) = 0.5$, $P(\bar{B}) = 0.7$ and $P(A \cup B) = 0.65$. Calculate the probability of the intersection $P(A \cap B)$ and determine whether the events are compatible.

A ball is drawn from an urn containing 10 balls numbered 1 to 10. Consider the events:

$A = \{ \text{drawing an even number} \}$

$B = \{ \text{drawing a multiple of 3} \}$

1. a) Write down the elements that make up the events $A \cup B$ and $A \cap B$.
2. b) Calculate $P(A \cup B)$ using Laplace's rule on the count of their elements.
3. c) Verify the previous result by applying the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Let $A$ and $B$ be two events in the same random experiment for which the following probabilities are known: $P(A) = 0.6$, $P(B) = 0.5$ and $P(\bar{A} \cap \bar{B}) = 0.2$.

Based on the properties of probability and De Morgan's laws, calculate:

1. a) The probability of the union of the events, $P(A \cup B)$.
2. b) The probability of the intersection, $P(A \cap B)$.
3. c) State with reasons whether the events $A$ and $B$ are compatible or incompatible.

In a study on digital content consumption among 16-year-olds, it was found that $70 \%$ use the platform StreamVibe and $45 \%$ use CloudMusic. Furthermore, it is known that $25 \%$ of young people do not use either of the two platforms. If a young person is chosen at random, calculate:

1. a) The probability that they use both platforms.
2. b) The probability that they use StreamVibe only.
3. c) The probability that they use CloudMusic but not StreamVibe.

We have the following partial contingency table showing the distribution of a group of 120 Sixth Form students according to their participation in extracurricular activities (Sport or Music):

1. a) Complete the contingency table in your notebook.
2. b) Calculate the probability that a randomly chosen student does not participate in either activity.
3. c) Are the events "Participating in Music" and "Participating in Sport" independent? Justify your answer mathematically.

A Year 13 student has prepared 18 of the 25 topics that make up the Geography syllabus. In the final exam, the teacher chooses two topics at random (without replacement). Calculate:

1. a) The probability that the student knows both topics.
2. b) The probability that they know neither.
3. c) The probability that they know at least one of the two selected topics.

Let $A$ and $B$ be two events such that $P(A) = 0.4$ and $P(B) = 0.3$. Calculate the probability of the union $P(A \cup B)$ under each of the following assumptions:

1. a) $A$ and $B$ are incompatible events.
2. b) $A$ and $B$ are independent events.
3. c) The event $B$ is contained in the event $A$ ($B \subset A$).

In a role-playing game, a player must pass two consecutive challenges. The probability of passing the first challenge is $0.7$. If they pass the first, the probability of passing the second is $0.8$, but if they fail the first, the probability of passing the second drops to $0.4$.

1. a) Draw the tree diagram representing this situation.
2. b) Calculate the probability that the player passes both challenges.
3. c) Calculate the probability that the player passes only one of the two challenges.

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

From a Spanish deck of 40 cards, two cards are drawn successively without replacing the first card in the deck. Determine the following probabilities:

1. a) Both cards are face cards (Jack, Knight, or King).
2. b) The first card is an Ace and the second is a King.
3. c) At least one of the two cards is Coins.

Consider the events $A$ and $B$ such that $P(A) = 0.5$, $P(B|A) = 0.4$ and $P(A \cup B) = 0.65$. From these data:

1. a) Find the probability of the intersection $P(A \cap B)$.
2. b) Calculate the value of $P(B)$.
3. c) Determine, giving reasons, whether the events $A$ and $B$ are independent.

On a streaming platform, it has been observed that $65\%$ of users under 20 watch animated series (event $A$) and $40\%$ watch nature documentaries (event $D$). Furthermore, it is known that the probability that a user does not watch animated series given that they do not watch nature documentaries is $0.5$.

Give reasoned answers to the following:

1. a) The probability that a user watches both types of content.
2. b) The probability that a user watches animated series but not nature documentaries.
3. c) Are the events $A$ and $D$ independent? Justify your answer using the previous results.

A video game development team is analysing the bugs in their latest release. They have classified the bugs into three categories: Critical ($C$), Graphics ($G$) and Sound ($S$). Following a mass test, the following data were obtained:

• $10\%$ of the bugs belong to all three types at the same time.

• The probability that a bug is Critical is $P(C) = 0.4$.

• The probability that it is Graphics is $P(G) = 0.5$.

• The probability that it is Sound is $P(S) = 0.35$.

• The probability that a bug is Critical and Graphics is $0.2$.

• The probability that it is Graphics and Sound is $0.15$.

• The probability that a bug does not belong to any of these three categories is $0.2$.

Calculate the probability that a randomly selected bug is:

1. a) Critical and Sound.
2. b) Exclusively a Graphics bug.
3. c) Sound, given that it is neither Critical nor Graphics.

Let $A$ and $B$ be two events in a random experiment such that their probabilities are $P(A) = x$ and $P(B) = y$. It is known that the union event has a probability of $P(A \cup B) = 0.75$ and that the intersection event satisfies $P(A \cap B) = 0.15$.

If it is also known that the probability of $A$ occurring given that $B$ does not occur is $P(A | \bar{B}) = 0.5$, determine the values of $x$ and $y$. Are $A$ and $B$ compatible events? Are they independent?

In an esports championship, the plays of a pro player are analysed. The events $K$ (getting a kill) and $W$ (winning the round) are defined. The analyst determines that:

• The probability of winning the round having achieved a kill is $P(W|K) = 0.8$.

• The probability of winning the round without having achieved any kills is $P(W|\bar{K}) = 0.3$.

• The probability that in a round the player wins or gets a kill is $0.85$.

Using event algebra, calculate:

1. a) The probability that the player gets a kill in a round, $P(K)$.
2. b) The probability that the player wins the round and does not get any kills.
3. c) Show whether the events "winning the round" and "getting a kill" are dependent or independent.

Using the properties of probability and De Morgan's laws, prove whether the following statement is true or false:

"If $A$ and $B$ are two mutually exclusive events with non-zero probabilities, then $P(\bar{A} | B) = 1$ always holds."

Support your reasoning with an original numerical example where $P(A) = 0.3$ and $P(B) = 0.4$.

A cybersecurity system has two independent protective barriers, $B_1$ and $B_2$. The probability that an external attack bypasses the first barrier is $P(B_1) = 0.15$ and the probability that it bypasses the second is $P(B_2) = 0.10$.

Calculate the probability that an attack:

1. a) Bypasses at least one of the two barriers.
2. b) Bypasses exactly one of the barriers.
3. c) Is not detected by either barrier, knowing that the attack has managed to bypass the first protection.

In a study on social media use, two 2nd year of Bachillerato students are selected consecutively and without replacement. It is known that $60\%$ of the school's students use the social network X.

If the probability that the second selected student uses the social network X, knowing that the first one uses it, is $35/59$, determine the total number of 2nd year Bachillerato students in that school.

Consider three events $A$, $B$ and $C$ such that $A$ and $B$ are independent, and $A$ and $C$ are mutually exclusive. The following probabilities are known:

$P(A) = 0.4$; $P(B) = 0.5$; $P(C) = 0.2$ and $P(B \cap C) = 0.1$.

Find the probability of the following compound events:

1. a) $P(A \cup B \cup C)$

1. b) $P((A \cap B) \cup C)$

1. c) $P(A \mid \bar{C})$