Exercises: Independence

Let $A$ and $B$ be two events of a random experiment. It is known that $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,2$. Determine analytically if 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 $P(C \cap D)$.

Events $E$ and $F$ are independent. Knowing that the probability of their intersection is $P(E \cap F) = 0,15$ and that $P(E) = 0,3$, find the probability of event $F$.

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 $P(H)$?

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

A card is drawn from a Spanish deck of $40$ cards, its suit is noted, it is returned to the deck, shuffled, and a second card is drawn. Calculate the probability that both drawn cards are of the "coins" suit.

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

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 an archery competition, archer Alpha has a probability of $0,8$ of hitting the center of the target, while archer Beta has a probability of $0,7$. If both shoot an arrow completely independently, calculate the probability that both hit the center of the target.

Two machines, $M_1$ and $M_2$, package dairy products in a factory and operate independently. The probability that machine $M_1$ suffers a breakdown 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 suffers a breakdown.

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

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

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 the same color.

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 hospital, a patient undergoes two independent diagnostic tests to detect a respiratory allergy. Test $X$ gives a false negative (it does not detect the allergy when the patient does have it) with a probability of $0,15$. Test $Y$ gives a false negative with a probability of $0,10$. If the tested patient is truly allergic, what is the probability that neither test 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 from 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 that operate completely independently. The probabilities that each sensor detects 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).

Consider two events $A$ and $B$ of the same sample space such that $P(A) = 0,5$, $P(B) = 0,4$ and $P(A \cup B) = 0,8$. Prove analytically that events $A$ and $B$ are not independent and, then, calculate the conditional probability $P(A|B)$.

An electronic circuit for a communications satellite is formed by three components $A$, $B$ and $C$ arranged as shown in the diagram. The circuit works correctly if component $A$ works AND, in addition, at least one of the 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 works correctly.

A language academy offers French and German classes. $60\%$ 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 academy's students study German.

A box contains $3$ green tokens (V) and $2$ orange tokens (N). Two consecutive draws are going to be made. Analyze the tree diagrams corresponding to performing the experiment with replacement (Panel A) and without replacement (Panel B). Calculate the probability of drawing a green token on the second draw in both cases and explain mathematically in which of the two scenarios the result of the second draw is independent of the result of the first.

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

In a Mendelian genetic cross, seed color (Yellow dominant, green recessive) and seed texture (Smooth dominant, wrinkled recessive) are inherited completely independently. If two heterozygous plants 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 green and wrinkled seeds simultaneously.

Two friends, Ashley and Chloe, alternately roll a fair six-sided die in a game. The first player to get a $6$ wins the game. If Ashley has the first turn and they roll independently, calculate the exact probability that Ashley wins the game on her first or second roll.