Exercises: Independence

Let $A$ and $B$ be two events in a random experiment. It is known that $P(A) = 0.4$, $P(B) = 0.5$ and $P(A \cap B) = 0.2$. Determine algebraically whether the 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. Given 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 in succession. Calculate the probability of obtaining "heads" on the first flip and "tails" on the second, assuming that the flips are independent events.

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

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

Cian is reviewing probability for the exam and comes across two events, $A$ and $B$, such that $P(A) = 0.3$, $P(B) = 0.5$, and $P(A \cap B) = 0$. In his notebook he writes: "Since the probability of the intersection is zero, it means that the two events have nothing to do with each other and, therefore, they are independent." Explain in detail what Cian's conceptual error is, prove analytically whether or not these events are independent by calculating $P(A) \times P(B)$, and state the general rule regarding the relationship between mutually exclusive events with positive probabilities and independence.

A cyclist rides daily along an avenue where they must pass through two traffic lights whose cycles are regulated completely independently. The probability of finding the first traffic light green is $P(S_1) = 0.7$ and that of finding the second one green is $P(S_2) = 0.6$.

1. a) Before performing any calculation, estimate with reasoning: will the probability of going through at least one of the two traffic lights on green ($P(S_1 \cup S_2)$) be greater than $0.7$, less than $0.6$, or between $0.6$ and $0.7$? Why?
2. b) Use the union formula for independent events to calculate the exact value of $P(S_1 \cup S_2)$ and check whether your initial estimation was correct, interpreting the result.

In an archery competition, female archer Alpha has a probability of $0.8$ of hitting the bullseye, while male archer Beta has a probability of $0.7$. If both shoot an arrow completely independently, calculate the probability that both hit the bullseye.

Two machines, $M_1$ and $M_2$, pack dairy products in a factory and operate independently. The probability of machine $M_1$ breaking down 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 breaks down.

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

In a statistical study on reading habits, $200$ people were surveyed. The results show that $120$ read regularly and $80$ do not. Furthermore, $90$ of the respondents have a university degree, of whom $54$ read regularly. Test mathematically whether the events "reading regularly" and "having a university 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 of the same colour.

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 (fails to detect the allergy when the patient actually has it) with a probability of $0.15$. Test $Y$ gives a false negative with a probability of $0.10$. If the tested patient is actually allergic, what is the probability that neither of the two tests 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 on 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 redundant security system with two independent sensors, $A$ and $B$, is installed in a house. The manufacturer guarantees that sensor $A$ has a probability of detecting an intruder of $P(A) = 0.8$. Security regulations require that the total probability of the alarm going off (that is, of at least one of the two sensors being triggered, $P(A \cup B)$) be exactly $0.96$. Knowing that both devices operate independently, set up the corresponding equation using the union formula and determine the minimum detection probability $P(B)$ that sensor $B$ must have to comply with said regulations.

Based on historical records from a weather station in the west of Ireland collected by Met Éireann during autumn, it is estimated that on any given day the probability of rain is $P(L) = 0.60$ and the probability of recording strong wind gusts is $P(V) = 0.40$. Likewise, the data reveal that it rains and is strongly windy at the same time on $30\,\%$ of the days ($P(L \cap V) = 0.30$).

1. a) Determine rigorously using the mathematical definition whether the events "rain" and "strong wind" are statistically independent in that region.
2. b) A resident argues: "Since storms in autumn bring wind and rain together, both phenomena cannot be independent." Explain whether the numerical result obtained in the previous section supports the resident's statement, comparing the actual overlap with what would be expected if they were independent.

A high-tech bank security system consists of three motion sensors operating completely independently. The probabilities of each sensor detecting 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 $B$ from the same sample space such that $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \cup B) = 0.8$. Show algebraically that events $A$ and $B$ are not independent, and then calculate the conditional probability $P(A|B)$.

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

At a language academy, French and German classes are offered. $60\%$ 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 students at the academy study German.

A box contains $3$ green counters (V) and $2$ orange counters (N). Two consecutive extractions are going to be carried out. Analyse the tree diagrams corresponding to carrying out the experiment with replacement (Panel A) and without replacement (Panel B). Calculate the probability of drawing a green counter on the second extraction in both cases, and explain mathematically in which of the two scenarios the result of the second extraction 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. To do so, use the algebraic properties of probability and the fundamental definition of independence.

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

Two friends, Aoife and Niamh, take turns rolling a fair six-sided die in a game. The winner is the first player to roll a $6$. If Aoife takes the first turn and they roll independently, calculate the exact probability that Aoife wins the game on her first or second roll.

The IT department of the Department of Education has 4 identical and independent backup servers to manage online enrollment applications. During opening day, the probability of a specific server crashing due to overload is $p = 0.15$. The overall system remains accessible as long as "at least one of the four servers remains operational".

To calculate the probability that the system remains accessible, two technicians propose different methods:

• Aoife suggests calculating separately and adding the probabilities that exactly 1, exactly 2, exactly 3, and all 4 servers work.

• Seán proposes calculating the probability that all 4 servers fail simultaneously and subtracting that value from $1$.

1. a) Compare both methods with reasoning, explaining which one is more advantageous and why in this case.
2. b) Apply the most efficient strategy to calculate the exact probability (to four decimal places) that the IT system does not crash.

In a pharmaceutical laboratory, two independent analytical tests, $T_1$ and $T_2$, are evaluated to identify a chemical compound in a sample. Test $T_1$ has a detection probability of $70\,\%$ ($P(T_1) = 0.7$), while test $T_2$ has an unknown probability $p = P(T_2)$.

The researcher Cian states in his preliminary report: "To achieve a probability of detecting the compound with at least one of the two tests of $95\,\%$, it is enough for $T_2$ to have a detection probability of $0.25$, since $0.70 + 0.25 = 0.95$".

1. a) Identify and explain the conceptual error Cian makes by ignoring the independence of the tests.
2. b) Set up the correct equation under the condition of independence and solve for the exact value of the probability $p = P(T_2)$ that is actually needed to achieve that $95\,\%$ combined effectiveness.
3. c) If the manufacturer of test $T_2$ only sells a standard version with $P(T_2) = 0.80$, calculate the maximum actual combined detection probability ($P(T_1 \cup T_2)$) that can be achieved by using both tests at the same time.