Dependent Events and Conditional Probability

In a class of $30$ students, $18$ study French, $12$ study German and $5$ study both languages. If a student is chosen at random and it turns out they study French, what is the probability that they also study German?

We roll two normal six-sided dice. Knowing that the sum of the scores obtained is $8$, what is the probability that a $3$ has been rolled on one of the dice?

In an urn there are $5$ red balls and $3$ blue balls. Two balls are drawn consecutively and without replacement. Calculate the probability that the first ball is red and the second is blue.

From a Spanish deck of $40$ cards, two cards are drawn at random one after the other and without replacement. What is the probability that both cards are kings?

In a factory, machine A produces $60\%$ of the parts and machine B produces the remaining $40\%$. It is known that $2\%$ of the parts produced by A are defective, while on machine B the percentage of defective parts is $5\%$. If a part is chosen at random from the warehouse, what is the probability that it was manufactured by machine B and is defective?

Observe the following tree diagram that represents a two-stage compound experiment. Calculate the probability of obtaining the final result $C$ passing through branch $A$.

We have two boxes. BOX I contains $4$ green balls and $2$ yellow balls. BOX II contains $3$ green balls and $5$ yellow balls. We draw a ball at random from BOX I and place it into BOX II. Then, we mix well and draw a ball from BOX II. What is the probability that both drawn balls are green?

From a box containing $10$ light bulbs, it is known that $3$ of them are blown. $3$ light bulbs are drawn successively and without replacement. What is the probability that exactly one of the drawn light bulbs is blown?

An opaque bag contains $10$ slips of paper identical to the touch: $2$ of them have a prize and $8$ are blank. Three friends, Ashley, Brandon, and Daniel, draw consecutively and without replacement one slip each in that same order. What is the probability that the first prize drawn is drawn by Daniel?

In an urn there are $x$ red balls and $4$ black balls. Two balls are drawn consecutively and without replacement. Knowing that the probability that both balls are red is exactly $\frac{1}{6}$, set up an equation and determine the number of red balls $x$ that are in the urn.

Consider two events $A$ and $B$ of a random experiment. We know that $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,2$. Check mathematically if events $A$ and $B$ are independent.

In a town, $45\%$ of the inhabitants read newspaper X, $35\%$ read newspaper Y, and $15\%$ read both newspapers. If we choose an inhabitant at random and we know that they read newspaper X, what is the probability that they also read newspaper Y? In light of the results, reason whether the events "reading newspaper X" and "reading newspaper Y" are independent.

Let $A$ and $B$ be two events of the same random experiment such that $P(A) = 0,5$, $P(B) = 0,4$ and $P(A \cup B) = 0,7$. Using the properties of probability, find the value of the intersection and calculate $P(A|B)$ and $P(B|A)$. Are $A$ and $B$ independent events? Reason your answer.

A coin is biased so that the probability of getting Heads is $0,7$. The coin is flipped twice consecutively. Calculate the probability of getting exactly one Heads and one Tails (in any order).

Two archers shoot an arrow at the same target. Archer A hits with a probability of $0,7$ and archer B hits with a probability of $0,8$. Their shots are completely independent. What is the probability that the target is hit by at least one arrow?

A computer security system has two independent firewalls. The probability that firewall A fails against an attack is $0,15$ and the probability that firewall B fails is $0,10$. The system is compromised only if both firewalls fail. What is the probability that the system resists an attack? If a third firewall C, identical to B and independent of the previous ones, is added, by how much does the probability that the system resists the attack increase?

A basketball player has a success rate of $80\%$ on the first free throw. If they make the first one, their confidence increases and the probability of making the second free throw goes up to $90\%$. However, if they miss the first one, they get nervous and the probability of making the second one drops to $60\%$. Calculate the probability that the player scores exactly one free throw in a series of two shots.

A rapid allergy test is performed in a clinic. It is known that $8\%$ of the patients are allergic to a certain substance. The test gives a positive result in $90\%$ of the patients who are truly allergic. However, it also gives a false positive in $5\%$ of the patients who are not allergic. A patient is chosen at random and the test is performed on them. What is the probability that the result is positive?

We have three chests that are identical on the outside. CHEST A contains $2$ gold coins and $1$ silver coins. CHEST B contains $1$ gold coin and $3$ silver coins. CHEST C contains $4$ gold coins. We roll a fair die: if it shows $1$ or $2$, we choose chest A; if it shows $3$, we choose chest B; if it shows $4$, $5$ or $6$, we choose chest C. Then, we draw a coin at random from the chosen chest. What is the probability that the drawn coin is gold?

On a bookshelf there are $5$ novels and $3$ poetry books. Two books are chosen at random one after the other and without replacement. If the second chosen book turned out to be poetry, what is the probability that the first chosen book was too?

In a high school, $55\%$ of the students are girls. It is known that $20\%$ of the girls and $25\%$ of the boys actively participate in the drama club. If a student who participates in the drama club is chosen at random, what is the probability that they are a girl?

We have two cubic dice in a shaker. The first is a normal die with faces numbered from $1$ to $6$. The second is a biased die that has three faces with a $2$, two faces with a $3$ and one face with a $5$. A die is chosen at random (with probability $\frac{1}{2}$ for each) and rolled. If the result obtained is an even number, what is the probability that the biased die was rolled?