Independence of Events

Let $A$ and $B$ be two events in a random experiment. It is known that $P(A) = 0.4$, $P(B) = 0.5$ and $P(A \cap B) = 0.2$. Determine algebraically whether the events $A$ and $B$ are independent.

In a random experiment, events $C$ and $D$ are independent. If $P(C) = 0.6$ and $P(D) = 0.3$, calculate the probability that both events occur at the same time, that is, find $P(C \cap D)$.

A fair coin is flipped twice in succession. Calculate the probability of obtaining "heads" on the first flip and "tails" on the second, assuming that the flips are independent events.

Observe the following tree diagram modelling two independent experiments. Calculate the probability that event $\overline{A}$ occurs in the first stage and event $B$ occurs in the second stage.

There are two urns with the composition shown in the figure. A ball is drawn at random from urn $A$ and, independently, another ball from urn $B$. Calculate the probability that both drawn balls are of the same color.

A box contains $3$ green counters (V) and $2$ orange counters (N). Two consecutive extractions are going to be carried out. Analyse the tree diagrams corresponding to carrying out the experiment with replacement (Panel A) and without replacement (Panel B). Calculate the probability of drawing a green counter on the second extraction in both cases, and explain mathematically in which of the two scenarios the result of the second extraction is independent of the result of the first.

Two events $M$ and $N$ are incompatible (mutually exclusive) and have probabilities strictly greater than zero. Can they be independent? Justify your answer using the mathematical definition of independence.

In a statistical study on reading habits, $200$ people were surveyed. The results show that $120$ read regularly and $80$ do not. Furthermore, $90$ of the respondents have a university degree, of whom $54$ read regularly. Test mathematically whether the events "reading regularly" and "having a university degree" are independent in this sample.

Consider two events $A$ and $B$ from the same sample space such that $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \cup B) = 0.8$. Show algebraically that events $A$ and $B$ are not independent, and then calculate the conditional probability $P(A|B)$.

Let $A$ and $B$ be two independent events associated with a random experiment. It is known that the probability of the union is $P(A \cup B) = 0.88$ and the probability of $A$ is $P(A) = 0.6$. Calculate the probability of event $B$.

In a coastal city, the probability of it raining on a Saturday is $40\%$, and the probability of it raining on a Sunday is $30\%$. Assuming that the weather on Sunday is independent of the weather on Saturday, find the probability that it rains during the weekend (that is, that it rains on Saturday, Sunday or both days).

Two machines, $M_1$ and $M_2$, pack dairy products in a factory and operate independently. The probability of machine $M_1$ breaking down on a production day is $0.05$, and that of machine $M_2$ is $0.08$. Calculate the probability that, on a given day, at least one of the two machines breaks down.

It is known that the probability of event $G$ occurring is $P(G) = 0.7$ and the probability of event $H$ occurring given that $G$ has occurred is $P(H|G) = 0.4$. If $G$ and $H$ are independent, what is the value of $P(H)$?

In a hospital, a patient undergoes two independent diagnostic tests to detect a respiratory allergy. Test $X$ gives a false negative (fails to detect the allergy when the patient actually has it) with a probability of $0.15$. Test $Y$ gives a false negative with a probability of $0.10$. If the tested patient is actually allergic, what is the probability that neither of the two tests detects their allergy?

A student has to take a multiple-choice exam. The probability that they know the correct answer to a History question is $0.7$, and the probability that they know the answer to a Geography question is $0.8$. If they are asked one question on each subject and their knowledge of them is independent, what is the probability that they know how to answer exactly one of the two questions?

A high-tech bank security system consists of three motion sensors operating completely independently. The probabilities of each sensor detecting an intruder entering the vault are $0.90$, $0.85$ and $0.95$, respectively. Calculate the probability that the system detects an intruder (that is, that at least one sensor is activated).

An electronic circuit for a communications satellite consists of three components $A$, $B$ and $C$ arranged as shown in the diagram. The circuit functions correctly if component $A$ works AND at least one of components $B$ or $C$ works. The three components fail independently with probabilities $P(F_A) = 0.02$, $P(F_B) = 0.1$ and $P(F_C) = 0.1$. Calculate the probability that the circuit as a whole functions correctly.

At a language academy, French and German classes are offered. $60\%$ of the students study French. It is known that the events "studying French" and "studying German" are completely independent. If the probability that a randomly chosen student studies neither of the two languages is $12\%$, determine what percentage of the students at the academy study German.

Prove analytically that if two events $A$ and $B$ are independent, then their complementary events $\overline{A}$ and $\overline{B}$ are also independent. To do so, use the algebraic properties of probability and the fundamental definition of independence.

In a Mendelian genetic cross, seed color (Yellow dominant, green recessive) and seed texture (Smooth dominant, wrinkled recessive) are inherited completely independently. If two plants that are heterozygous for both traits are crossed, the probability of inheriting the recessive allele from each parent for a single gene is $0.5$. Calculate the probability that a randomly chosen offspring plant has seeds that are both green and wrinkled.

Two friends, Carmen and Daniela, take turns rolling a fair six-sided die in a game. The winner is the first player to roll a $6$. If Carmen takes the first turn and they roll independently, calculate the exact probability that Carmen wins the game on her first or second roll.