Exercises: Dependence

An urn contains $5$ 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?

Consider two events $A$ and $B$ of a random experiment. It is known that $P(A) = 0.4$, $P(B) = 0.5$ and $P(A \cap B) = 0.2$. Check mathematically whether the events $A$ and $B$ are independent.

We roll two standard six-sided dice. Given that the sum of the scores obtained is $8$, what is the probability that a $3$ was rolled on one of the dice?

Look at the following tree diagram representing a two-stage compound experiment. Calculate the probability of obtaining the final result $C$ going through branch $A$.

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 turns out to study French, what is the probability that they also study German?

A card is drawn from a Spanish deck of cards, and then a coin is tossed. Are these two events dependent or independent? Calculate the probability of getting an "Ace" from the deck and "Heads" on the coin.

A coin is biased such that the probability of getting Heads is $0.7$. The coin is tossed twice consecutively. Calculate the probability of getting exactly one Head and one Tail (in any order).

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 for 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?

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

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 know that they read newspaper X, what is the probability that they also read newspaper Y? In light of the results, explain whether the events "reading newspaper X" and "reading newspaper Y" are independent.

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 without replacement. What is the probability that exactly one of the drawn bulbs is blown?

A rapid allergy test is carried out 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 actually allergic. However, it also gives a false positive in $5\%$ of the patients who are not allergic. A patient is chosen at random and tested. What is the probability that the result is positive?

Two archers shoot an arrow at the same target. Archer A hits the target with a probability of $0.7$ and Archer B hits the target 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?

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

We have three chests that are externally identical. 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 we get $1$ or $2$, we choose chest A; if we get $3$, we choose chest B; if we get $4$, $5$ 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?

In a 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?

In an urn there are $x$ red balls and $5$ 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 $x$ in the urn.

A computer security system has two independent firewalls. The probability that firewall A fails during 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 withstands an attack? If a third firewall C, identical to B and independent of the previous ones, is added, by how much does the probability of the system withstanding the attack increase?

A player draws cards from a Spanish deck of $40$ cards, one by one and without replacement, until a face card (jack, knight, or king) is drawn. What is the probability that they have to draw exactly $3$ cards to achieve their goal?

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? Explain your answer.

An opaque bag contains $10$ tickets that feel identical to the touch: $2$ of them are winning tickets and $8$ are blank. Three friends, Fátima, Bruno and Ionut, consecutively draw one ticket each, without replacement, in that order. What is the probability that the first winning ticket is drawn by Ionut?

We have two cubic dice in a shaker. The first is a standard 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 a probability of $\frac{1}{2}$ for each) and rolled. If the score obtained is an even number, what is the probability that the biased die was rolled?