Properties of Probabilities

A video game developer is analysing the probability of obtaining rare items from a loot box. The possible outcomes are: "Common Item" ($C$), "Rare Item" ($R$) and "Epic Item" ($E$). State whether the following probability assignment is valid according to Kolmogorov's axioms and justify your answer:

$P(C) = 0.65$

$P(R) = 0.20$

$P(E) = 0.25$

In a study on mobile app usage among teenagers, it has been determined that the probability of a student having a video editing app installed is $P(V) = 0.35$. Calculate the probability that a student chosen at random does not have this app installed.

In a box of materials for a technology project there are 50 resistors. It is known that some are $100\ \Omega$ ($A$) and others are $220\ \Omega$ ($B$), and some are new and others used. One resistor is drawn at random. If $P(A) = 0.6$ and events $A$ and $B$ are complementary, determine:

1. a) The probability of drawing a $220\ \Omega$ resistor.
2. b) What is the value of $P(A \cap B)$? Justify your answer based on the relationship between complementary events.

For the graduation trip, Year 13 students must choose between three possible destinations: Mojácar, Mallorca or Tenerife. It is known that the probability of them choosing Mojácar is $P(M) = 0.42$ and that of choosing Mallorca is $P(L) = 0.38$. Given that the events are mutually exclusive (they can only choose one destination), calculate:

1. a) The probability that the students travel to Mojácar or Mallorca.
2. b) The probability that they choose neither of these two destinations and go to Tenerife.

Let $X$ and $Y$ be two events such that $P(X) = 0.25$ and $P(Y) = 0.55$. Given that these events are mutually exclusive, calculate the probability that:

1. a) Event $X$ or event $Y$ occurs.
2. b) Event $X$ does not occur.
3. c) $X$ and $Y$ occur at the same time.

According to a school's statistics, $75\,\%$ of the students regularly use Instagram ($I$), $60\,\%$ use TikTok ($T$) and $45\,\%$ use both social networks at the same time.

Calculate the probability that a randomly selected student uses Instagram or TikTok, that is, $P(I \cup T)$.

From the data represented in the following Venn diagram, where the values indicate the probabilities of each region, calculate the required probabilities:

1. a) $P(A)$
2. b) $P(B)$
3. c) $P(A \cup B)$

Consider events $A$ and $B$ associated with a random experiment. If we know that $P(A) = 0.5$, $P(B) = 0.4$ and that the probability of neither $A$ nor $B$ occurring is $P(\bar{A} \cap \bar{B}) = 0.3$, apply de Morgan's Laws to find the probability of the union, $P(A \cup B)$.

Let $A$ and $B$ be two events associated with a random experiment. It is known that the probability that neither of them occurs is $P(\bar{A} \cap \bar{B}) = 0.2$. Furthermore, it is known that $P(A) = 0.5$ and $P(B) = 0.4$.

Calculate, giving reasons, the following probabilities:

1. a) $P(A \cup B)$
2. b) $P(A \cap B)$
3. c) $P(\bar{A} | B)$
4. d) Determine whether events $A$ and $B$ are independent.

On a streaming platform, $60 \%$ of young subscribers watch sci-fi series (event $S$), $45 \%$ watch nature documentaries (event $D$) and $20 \%$ of users watch both types of content.

Calculate the probability that a user chosen at random:

1. a) Watches sci-fi series or nature documentaries.
2. b) Watches neither of the two genres.
3. c) Watches only one of the two genres.
4. d) Given that they do not watch sci-fi series, what is the probability that they watch nature documentaries?

Observe the following Venn diagram representing the probabilities of different regions for two events $M$ and $N$ in a sample space $E$.

From the data in the diagram, calculate the following probabilities:

1. a) $P(M \cup N)$
2. b) $P(M - N)$, that is, the probability of $M$ occurring but not $N$.
3. c) $P(\bar{M} \cup \bar{N})$ using de Morgan's Laws.
4. d) Are events $M$ and $N$ mutually exclusive? Justify your answer.

In a study on social media use among Year 13 students, two platforms are analysed: InstaPlus (event $A$) and TikToker (event $B$). It is known that the probability of a student using at least one of the two is $0.8$. Furthermore, the probability of using the first but not the second is $0.3$. If it is known that the probability of using both platforms is half the probability of using neither of them, calculate:

1. a) The probability that a student, chosen at random, uses the TikToker platform.

1. b) Are events $A$ and $B$ independent? Justify your answer mathematically by comparing $P(A \cap B)$ with the product of their individual probabilities.

1. c) Calculate the probability that they use only one of the two platforms.

In a study on sustainable mobility among Sixth Form students, it was found that the probability of a student using a bicycle to get to school is $P(B) = 0.25$, that of using public transport is $P(T) = 0.60$ and that of using neither of these two modes of transport is $P(\bar{B} \cap \bar{T}) = 0.30$.

1. a) Are the events "using a bicycle" and "using public transport" compatible?
2. b) Calculate the probability that a student uses only a bicycle.
3. c) What is the probability that they use public transport given that they do not use a bicycle?

In a university entrance exam, $75\%$ of students pass Mathematics ($M$) and $60\%$ pass History ($H$). It is known that $10\%$ of students pass neither of the two subjects.

1. a) Calculate the probability that a student passes both subjects.

1. b) Given that a student has passed History, what is the probability that they have also passed Mathematics?

1. c) What is the probability that a student passes Mathematics but not History? Explain the relationship between this result and the difference of events $P(M - H)$.

An eSports team is analysing its performance on two different maps in a video game: Inferno (event $I$) and Mirage (event $M$). Statistics show that they win on Inferno with a probability of $0.7$ and that the probability of winning on Mirage is $0.5$. It has been verified that the probability of winning on at least one of the two maps is $0.85$.

1. a) Calculate the probability that they win on both maps.
2. b) Is winning on Inferno independent of winning on Mirage?
3. c) Calculate the probability that they win on Mirage given that they lost on Inferno.

Given two events $A$ and $B$ from the same random experiment, it is known that:

• The probability of $A$ occurring is three times that of its complement, $\bar{A}$.

• $P(B) = 0.4$.

• $P(A \cap B) = 0.3$.

Calculate, giving reasons:

1. a) $P(A)$.
2. b) $P(A \cup B)$.
3. c) $P(\bar{A} \cap B)$.
4. d) $P(\bar{B} | \bar{A})$.

Consider two independent events $X$ and $Y$. It is known that $P(X) = 0.6$ and $P(X \cup Y) = 0.8$. Using the properties of probability and the definition of independence, calculate:

1. a) The probability of event $Y$.
2. b) The probability that $X$ occurs but $Y$ does not occur.
3. c) The probability that neither of the two occurs.

A task management app classifies notifications according to their urgency ($U$) and their type: personal ($P$) or work ($T$). It is known that $P(U) = 0.35$, $P(P) = 0.55$ and $P(U \cup P) = 0.75$.

Determine:

1. a) Whether events $U$ and $P$ are independent.
2. b) The probability that a notification is urgent given that it is personal.
3. c) The probability that a notification is personal but not urgent.
4. d) If notifications are either personal ($P$) or work ($T$), calculate the probability that a notification is not urgent and is for work, that is, $P(\bar{U} \cap T)$.

Let $A$ and $B$ be two events of a random experiment such that their probabilities are given in terms of a parameter $k$. It is known that $P(A) = k$, $P(B) = 0.5$ and $P(A \cup B) = 0.7$.

1. a) Determine the value of $k$ for events $A$ and $B$ to be mutually exclusive.

1. b) Determine the value of $k$ for events $A$ and $B$ to be independent.

1. c) If $k = 0.4$, calculate the probability of $B$ occurring given that $A$ has not occurred, that is, $P(B | \bar{A})$.

A cybersecurity system monitors two types of attacks: Phishing ($A$) and Malware ($B$). The metrics for the last month indicate that $P(\bar{A} \cap \bar{B}) = 0.15$ and $P(A \cap B) = 0.25$. If the probability of suffering a Phishing attack is double that of suffering a Malware attack ($P(A) = 2P(B)$), find:

1. a) The probability of suffering a Malware attack.

1. b) The probability that, having suffered a Malware attack, it is not a Phishing attack.

1. c) Using de Morgan's Laws, demonstrate whether the union of the complements $P(\bar{A} \cup \bar{B})$ or the complement of the intersection $\overline{P(A \cap B)}$ is more likely.

Let $A$, $B$ and $C$ be three events such that $A$ and $B$ are independent of each other, and $C$ is an event mutually exclusive with $A$ and with $B$. The following probabilities are known: $P(A) = 0.4$, $P(B) = 0.3$ and $P(C) = 0.1$. Calculate, giving reasons:

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

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

1. c) $P(\bar{A} | \bar{B})$

In an eSports tournament, a player has a probability $p$ of winning the first match. If they win the first match, their confidence increases and the probability of winning the second match is $0.7$. If they lose the first match, the probability of winning the second match drops to $0.3$. Given that the probability of winning the second match is $0.54$, calculate:

1. a) The value of the probability $p$.

1. b) The probability that the player won the first match given that they lost the second match.

A music streaming app classifies its users into three categories according to their listening time: Free ($F$), Premium ($P$) and Family ($S$). $50\%$ are Free users, $30\%$ are Premium users and the rest are Family users. It is known that the percentage of users who regularly listen to podcasts is $20\%$ in the Free category, $60\%$ in the Premium category and $40\%$ in the Family category.

1. a) If we choose a user at random who listens to podcasts, what is the probability that they belong to the Premium plan?

1. b) If a new condition is added: "No Family user who listens to podcasts is under 18 years old". If we choose a user at random, what is the probability that they are a Family user, listen to podcasts and are under 18 years old? Justify your answer using the impossible event.

Formally prove, using Kolmogorov's axioms and the properties of probability, the following statement: "If $A$ and $B$ are two independent events, then their complements $\bar{A}$ and $\bar{B}$ are also independent".

Hint: You must start from the definition of independence $P(A \cap B) = P(A) \times P(B)$ and arrive at $P(\bar{A} \cap \bar{B}) = P(\bar{A}) \times P(\bar{B})$.