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 pack of $40$ cards (four suits of ten cards each), 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 pack 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).

Cian has $5$ black socks and $3$ blue socks in a drawer, all loose and identical to the touch. He prepares to draw two socks consecutively, in the dark and without replacement. To calculate the probability that the first sock is black and the second is blue, Cian writes in his notebook:

"$P(\text{Black on 1st}) = \frac{5}{8}$ and $P(\text{Blue on 2nd}) = \frac{3}{8}$. Since they are two independent events, I multiply both probabilities: $P(\text{Black and Blue}) = \frac{5}{8} \times \frac{3}{8} = \frac{15}{64}$".

1. a) Point out the conceptual error Cian made and explain in your own words why two consecutive draws without replacement cannot be treated as independent events.
2. b) Correct the calculation by applying the multiplication rule for dependent events: $P(A \cap B) = P(A) \times P(B \mid A)$.
3. c) Calculate the probability that the two drawn socks are of the same color.

In a group of $30$ 4th Year students, a survey on hobbies is conducted: $18$ regularly play video games, $12$ participate in club sports, and $6$ do both activities.

1. a) If a student is chosen at random from the entire class, calculate the prior probability that they do club sports, $P(D)$.
2. b) Before doing any formal calculation, make a reasoned estimate: if we randomly choose a student only from among those who play video games, will the probability that they do club sports be greater than, equal to, or less than the one obtained in the previous part? Justify your prediction by comparing the overall proportion with that of the subgroup.
3. c) Numerically verify your estimate by calculating the conditional probability $P(D \mid V)$ using its definition $P(D \mid V) = \frac{P(D \cap V)}{P(V)}$. Does the exact result confirm your previous reasoning?

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?

According to mobility data inspired by the Road Safety Authority (RSA) for a holiday travel operation, in a county $70\%$ of intercity trips are made by motorway ($A$) and the remaining $30\%$ by conventional secondary roads ($S$). It is known that the probability of experiencing a severe traffic jam ($R$) is $0.15$ when travelling on a motorway, whereas on secondary roads this probability drops to $0.05$.

1. a) Model the situation by constructing a complete tree diagram that shows the events of each stage and their associated probabilities (prior and conditional).
2. b) Use the law of total probability to calculate the probability that a driver chosen at random on that day experiences a severe traffic jam.
3. c) If a driver reaches their destination after having experienced a severe traffic jam, what is the probability that they made their trip on a motorway? Explain why this percentage is so high even though motorways are designed to carry smoother traffic flow.

In an opaque urn there are $10$ balls identical to the touch, of which $r$ are red and the rest are white. Two balls are drawn at random, consecutively and without replacement.

1. a) Knowing that the probability that both drawn balls are red is exactly equal to $\frac{1}{15}$, set up an algebraic equation in terms of $r$ using the multiplication rule for dependent events and solve it to find how many balls of each color the urn contained at the beginning.
2. b) With the composition of the urn now deduced, calculate the probability that the second ball is white knowing that the first ball drawn was red, $P(B_2 \mid R_1)$.
3. c) Mathematically prove whether the event "getting a white ball on the second draw" is dependent on or independent of the result obtained on the first, by comparing $P(B_2 \mid R_1)$ with the prior probability of getting a white ball on the first draw, $P(B_1)$.

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 pack of $40$ cards (four suits of ten cards each, with a jack, knight and king in each suit), 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, Aoife, Seán and Niamh, consecutively draw one ticket each, without replacement, in that order. What is the probability that the first winning ticket is drawn by Niamh?

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?

In a school health center, the reliability of a rapid diagnostic test to detect an asymptomatic infection is being evaluated. Epidemiological data indicate that the prevalence of the infection in the target population is $2\%$ ($0.02$). The sensitivity of the test is $95\%$ (if a person is infected, they test positive with probability $0.95$) and the false positive rate is $5\%$ (if a person is healthy, they test positive by mistake with probability $0.05$). We want to calculate the probability that a person with a positive result is truly infected, $P(I \mid +)$.

Two students propose different solving methods:

• Method 1 (Analytical formal): Define events, calculate the total probability $P(+)$ by adding the two branches of the tree, and apply Bayes' theorem: $P(I \mid +) = \frac{P(I \cap +)}{P(+)}$.

• Method 2 (Contingency table with natural frequencies): Assume a hypothetical population of $10,000$ people, distribute them into a $2 \times 2$ two-way table according to their true status and the test, and calculate the direct ratio of infected people to the total number of positives.

1. a) Solve the problem using both methods and verify that you reach exactly the same numerical value (round the result to one decimal place as a percentage).
2. b) Critically compare both approaches: which of the two methods do you consider more suitable and understandable for explaining to a patient without mathematical training why a test with "$95\%$ accuracy" yields such an unexpectedly low real probability of being sick? Justify your choice by explaining the role played by false positives in the denominator.

In a criminal case, a biological sample found at the crime scene is analyzed. The forensic report concludes that only $1$ in every $100,000$ randomly chosen people shares this genetic profile; that is, the probability of an accidental match in an innocent individual is $P(\text{Match} \mid \text{Innocent}) = 0.00001$.

The trial prosecutor accuses the suspect Cian, arguing the following: "Given that Cian's DNA fully matches the sample, the probability that he is innocent is barely $0.001\%$, so his guilt is proven with $99.999\%$ certainty."

The judge requests an expert report assuming that the crime was committed in a closed city of $500,000$ adult inhabitants, among whom the sole culprit is found, and knowing that Cian was identified solely because he resides in the city and underwent general screening, with no further evidence against him.

1. a) Identify the serious conceptual error made by the prosecutor (known as the "prosecutor's fallacy"), clearly explaining which two distinct conditional probabilities he has confused and writing each with its symbolic notation $P(\dots \mid \dots)$.
2. b) Using Bayes' theorem or deducing the expected number of matches in the population of $500,000$ inhabitants, calculate the true probability that Cian is guilty knowing that his DNA has matched: $P(\text{Guilty} \mid \text{Match})$.
3. c) Write a conclusion addressed to the court justifying, based on the numerical result obtained, why the DNA test alone is insufficient to convict the accused without additional evidence.