Properties of probability

An urn contains numbered balls. Let the events be $A$ = "drawing an even number" and $B$ = "drawing an odd number less than 5". If we know that $P(A) = 0,5$, $P(B) = 0,2$ and that the events are mutually exclusive, calculate the probability of the union $P(A \cup B)$.

A study on digital habits reveals that the probability of a teenager using a microblogging social network is $P(S) = 0,72$.

Calculate the probability of the complementary event, that is, the probability that a teenager chosen at random does not use that social network.

Using De Morgan's Laws, calculate the probability that neither event $A$ nor event $B$ occurs (that is, $P(\bar{A} \cap \bar{B})$), knowing that the probability of the union is $P(A \cup B) = 0,84$.

On a streaming platform, $65\ \%$ of young users are subscribed to the Music Plan ($M$) and $40\ \%$ are subscribed to the Video Plan ($V$). It is also known that $25\ \%$ of users enjoy both plans simultaneously.

Calculate the probability that a user chosen at random is subscribed to at least one of the two plans (music or video).

Let $A$ and $B$ be two events such that $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cup B) = 0,7$. Determine whether the events $A$ and $B$ are compatible or mutually exclusive, justifying your answer by calculating $P(A \cap B)$.

In a study on digital content consumption in a Year 13 class, it is known that the probability of a student regularly using TikTok (event $A$) is $0,65$, the probability of using Instagram (event $B$) is $0,70$ and the probability of using neither of the two social networks is $0,15$.

Using the properties of probability, calculate:

1. a) The probability that a student, chosen at random, uses both social networks.
2. b) The probability that they use only one of the two.
3. c) Are the events $A$ and $B$ independent? Justify your answer.

At a sports center, $60\%$ of the members do strength training (event $F$) and $45\%$ do yoga (event $Y$). It is also known that $20\%$ of those who do strength training are also enrolled in yoga.

Construct a contingency table or a Venn diagram to answer the following questions:

1. a) Probability that a randomly chosen member does neither strength training nor yoga.

1. b) If we know that a member does yoga, what is the probability that they do not do strength training?

1. c) Are the events "doing strength training" and "doing yoga" independent?

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

Consider two events $A$ and $B$ of a random experiment. It is known that $P(A) = 0,6$ and $P(B) = 0,7$.

Based on Kolmogorov's axioms, prove that:

1. The minimum possible value for $P(A \cap B)$ is $0,3$.

2. The maximum possible value for $P(A \cap B)$ is $0,6$.

3. If a third event $C$ is added such that $P(C) = 0,8$, what is the minimum value of the probability that all three events occur simultaneously, $P(A \cap B \cap C)$? Justify your answer using Bonferroni's inequality: $P(A \cap B) \geq P(A) + P(B) - 1$.