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.

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.

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.

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.