Exercises: Independent events

A fair coin is flipped and a standard cubic die (with faces numbered from 1 to 6) is rolled. What is the probability of obtaining "Tails" on the coin and a "5" on the die?

Two events $A$ and $B$ of a random experiment satisfy $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,2$. Are events $A$ and $B$ independent? Justify your answer mathematically.

In an urn there are 3 red balls and 4 blue balls. A ball is drawn, its colour recorded, returned to the urn and a second ball is drawn. Calculate the probability that the first ball is red and the second is blue.

Two archers, Ana and Carlos, shoot at a target. The probability that Ana hits the bullseye is $0,7$ and that Carlos hits it is $0,6$. If both shoot independently, what is the probability that both hit the bullseye?

On a European roulette wheel there are 18 red pockets, 18 black pockets and 1 green pocket (zero). If the roulette wheel is spun twice consecutively, what is the probability that the ball lands in a black pocket on both spins?

It is known that two events $A$ and $B$ are independent. If the probability that $A$ occurs is $P(A) = 0,3$ and the probability that $B$ occurs is $P(B) = 0,8$, calculate the probability that at least one of the two occurs, that is, $P(A \cup B)$.

A student answers a short 3-question test completely at random. Each question has 4 answer options and only one is correct. What is the probability of answering all the questions correctly?

From a Spanish deck of 40 cards, a card is drawn, its value recorded, replaced in the deck and a second card is drawn. Find the probability of drawing a "King" on the first draw and a card of the suit of "Cups" on the second.

In a 4th-year ESO class, Sara and her group conduct an experiment with an urn containing 4 red balls and 6 black balls. The experiment consists of drawing a ball, noting its colour, returning it to the urn (sampling with replacement), and drawing a second ball.

Sara states the following: "Since we return the ball, the two draws are independent events. The probability of drawing red on the first draw is $\frac{4}{10} = 0,4$ and on the second draw it is also $0,4$. Therefore, the probability of drawing two red balls is $0,4 + 0,4 = 0,8$, because since they are independent, you just have to add the probabilities."

1. a) Identify and explain, with reasoning, Sara's conceptual mistake. Why does adding probabilities lack logical sense here (think about what would happen if she did 3 draws)?
2. b) Correct the calculation by applying the proper property of independent events to obtain the exact probability of drawing two consecutive red balls.

According to punctuality data published for the Renfe Cercanías network in Madrid, the probability that a morning train on a given line arrives on time at its destination is $0,92$ ($92\,\%$). It is assumed that technical incidents and delays on different working days act as independent events. A 4th-year ESO student takes that train on Monday and Tuesday to go to school.

1. a) Justify what it means in this real-life context for punctuality on Monday and on Tuesday to be "independent" events.
2. b) Mathematically model the situation and calculate the probability that the train is on time on both days.
3. c) Calculate the probability that the train is delayed on at least one of the two days.

In a factory, two quality control machines inspect products independently. Machine 1 detects a defect with a probability of $0,9$, while machine 2 does so with a probability of $0,85$. If a defective product passes through both machines, what is the probability that at least one of the two detects the defect?

In a population, the probability of having the genotype for blue eyes is $0,25$ and the probability of having the genotype for blond hair is $0,15$. Assuming that the inheritance of both traits is independent, calculate the probability that a person chosen at random has exactly one of the two traits.

A basketball player has a free throw success rate of $80\%$. If they take 3 free throws independently, what is the probability that they score exactly two of them?

A hospital computer system has 3 independent servers to prevent service downtime. The probability of a server failing on any given day is $0,05$. What is the probability that the system keeps working (that is, at least one server does not fail) for an entire day?

Let $A$ and $B$ be two independent events of a random experiment. It is known that the probability of the union is $P(A \cup B) = 0,85$ and that $P(A) = 0,6$. Calculate the value of the probability of event $B$, that is, $P(B)$.

The probability that it rains on an autumn Monday is $0,3$. Independently, the probability that the school bus arrives late on that same day is $0,2$. What is the probability that next Monday it does not rain and the bus arrives late?

A jeweller's shop has two independent alarm systems. During an attempted robbery, the main alarm sounds with a probability of $0,95$, and the secondary alarm sounds with a probability of $0,90$. What is the probability that during a robbery only one of the two alarms sounds?

A student sits two language exams: English and French. The probability of passing English is $0,7$. It is known that the probability of passing both exams is $0,56$. Assuming that the results of both tests are independent, what is the probability that she fails the French exam?

On the theoretical driving test for obtaining a driving licence from the Dirección General de Tráfico (DGT), many questions are multiple-choice with 3 possible answers ($A$, $B$, and $C$), of which only one is correct. Imagine a person who has not studied the material and answers a block of 4 consecutive questions completely at random. The answer chosen for each question is independent of the others.

1. a) Estimate: Without performing detailed calculations with a calculator, do you think the probability of getting all 4 questions right purely by chance will be higher than $5\,\%$, or do you think it will be below $2\,\%$? Write down your initial intuition and briefly explain what you base it on.
2. b) Verify: Apply the product rule for independent events and calculate the exact value of the probability of getting all 4 questions right in a row. Express the result as an irreducible fraction, as a decimal number, and as a percentage rounded to two decimal places.
3. c) Compare the value obtained with your previous estimate and assess whether it is reasonable to rely on chance in a multiple-choice test.

Based on historical climatological series from the State Meteorological Agency (AEMET) for the month of November in a northern mainland town like Santander, the probability that any given day dawns raining is $0,45$ ($45\,\%$). To organize a weekend outing (Saturday and Sunday), a scout group models the weather assuming that Saturday's weather and Sunday's weather behave as independent events.

1. a) Model the situation and calculate the probability that the scout group can carry out their outdoor activities without rain on either of the two days.
2. b) Calculate the probability that it rains on at least one of the two days of the weekend.
3. c) Reason critically: in real meteorology, if an Atlantic storm hits the Cantabrian coast on Saturday, is it realistic to assume that Sunday's weather is mathematically independent of Saturday's? Explain how this affects the reliability of the mathematical model.

A four-engine cargo plane has 4 engines operating completely independently. The probability of an engine failing during a transoceanic flight is $p = 0,01$. The plane can complete its flight safely if at least 2 of its engines function correctly. What is the probability that the plane suffers an accident due to multiple engine failure?

Two events $A$ and $B$ are independent. It is known that the probability that at least one of them occurs is $P(A \cup B) = 0,76$ and the probability that both occur simultaneously is $P(A \cap B) = 0,24$. Find the exact values of $P(A)$ and $P(B)$, knowing that event $A$ is more likely than event $B$.

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

Two friends, Santiago and Diana, roll a fair die alternately. The first person to roll a "6" wins the game and the game ends. If Santiago rolls first, calculate the exact probability that Santiago wins the game. (Hint: consider the turns as an infinite geometric series).

Let $A$, $B$ and $C$ be three mutually independent events in a laboratory experiment. Their probabilities of success are $P(A) = 0,5$, $P(B) = 0,4$ and $P(C) = 0,6$. Calculate the probability that exactly two of the three events occur.

An Olympic archer hits the exact centre of the target (bullseye) with a probability of $p = 0,2$ on each shot. The shots are independent events. What is the minimum number of arrows they must shoot so that the probability of hitting the centre at least once is greater than $0,99$?

Urn A contains $x$ white balls and $4$ black balls. Urn B contains $3$ white balls and $y$ black balls. One ball is drawn independently from each urn. Given that the probability that both balls are white is $\frac{1}{8}$ and the probability that both are black is $\frac{5}{12}$, determine the number of white balls in Urn A ($x$) and the number of black balls in Urn B ($y$).

An electronic circuit consists of three components, $A$, $B$ and $C$, which operate independently. Component $A$ is connected in series with a block formed by $B$ and $C$ in parallel. The probabilities of each component functioning correctly without failing are $P(A) = 0,9$, $P(B) = 0,8$ and $P(C) = 0,7$. Calculate the probability that electric current can pass from one end of the circuit to the other.

A small e-commerce company in Valencia stores backups of its orders on three identical, independent external hard drives. Due to wear and tear and continuous use, the probability of a hard drive failing overnight is $0,05$ (and $0,95$ that it operates correctly). For the company to be able to operate the next day without losing data, it is sufficient that at least one of the three drives works.

To calculate the probability that the system keeps the data safe, two technicians propose different approaches:

• Laura's method (direct): Calculate and add separately the probability that exactly one drive works (3 possible cases), exactly two drives work (3 possible cases), and all three drives work (1 case).

• Carlos's method (complementary event): Calculate the probability that all three drives fail at the same time and subtract that value from 1.

1. a) Compare both methods and justify which of the two is preferable in this problem and why.
2. b) Apply the most efficient method to calculate the exact probability that the company preserves its backups that night.
3. c) If the company were to grow and have 8 independent hard drives instead of 3, which of the two methods would maintain a similar calculation difficulty and which would become virtually unmanageable by hand? Justify your answer.

In a care home in Andalusia, safety regulations require installing autonomous smoke detectors in bedroom hallways. The detectors are battery-operated completely independently of each other. The manufacturer's technical specifications state that each individual detector has a correct activation probability of $0,80$ in the actual presence of smoke (that is, a failure probability of $0,20$).

The management of the centre requires an overall safety guarantee: the probability that at least one of the detectors placed in a hallway triggers the alarm in the event of an incipient fire must be at least $99\,\%$ ($0,99$).

1. a) If $n$ independent detectors are installed in the hallway, write the algebraic expression (in terms of $n$) representing the probability that none of the detectors work in the presence of smoke.
2. b) Set up the inequality corresponding to the safety condition: $P(\text{at least one works}) \ge 0,99$.
3. c) Through successive checks for $n = 1, 2, 3, \dots$, determine the minimum number of independent detectors that must be placed in each hallway to meet the safety requirement and interpret the result.