Exercises: Properties of probability

On a streaming platform, $65\ \%$ of young users are subscribed to the Music Plan ($M$) and $40\ \%$ are subscribed to the Video Plan ($V$). It is also known that $25\ \%$ of users enjoy both plans simultaneously.

Calculate the probability that a user chosen at random is subscribed to at least one of the two plans (music or video).

Observe the following Venn diagram representing the events $A$ (students who play basketball) and $B$ (students who do swimming) at a sports centre:

Knowing that the numbers represent the probability of each region, use the properties of probability to calculate:

1. a) $P(A)$
2. b) $P(A \cap B)$
3. c) $P(\bar{A})$ (Probability of not playing basketball).

A study on digital habits reveals that the probability of a teenager using a microblogging social network is $P(S) = 0.72$.

Calculate the probability of the complementary event, that is, the probability that a teenager chosen at random does not use that social network.

Let $A$ and $B$ be two events such that $P(A) = 0.4$, $P(B) = 0.5$ and $P(A \cup B) = 0.7$. Determine whether the events $A$ and $B$ are compatible or mutually exclusive, justifying your answer by calculating $P(A \cap B)$.

In an e-sports competition, the probability that team "Alfa" wins the first game is $P(A) = 0.6$. The probability that it wins the second is $P(B) = 0.5$. If the probability that it wins both is $P(A \cap B) = 0.3$, calculate:

1. a) The probability that it wins the first game but not the second ($P(A \cap \bar{B})$).

1. b) Illustrate the situation using an area diagram like the following one, in which the whole square represents probability 1 and each region has an area equal to its probability. Fill in the probability of the other three regions.

Using De Morgan's Laws, calculate the probability that neither event $A$ nor event $B$ occurs (that is, $P(\bar{A} \cap \bar{B})$), knowing that the probability of the union is $P(A \cup B) = 0.84$.

An urn contains numbered balls. Let the events be $A$ = "drawing an even number" and $B$ = "drawing an odd number less than 5". If we know that $P(A) = 0.5$, $P(B) = 0.2$ and that the events are mutually exclusive, calculate the probability of the union $P(A \cup B)$.

Given two events $A$ and $B$ of a random experiment, the following data are known:

$P(\bar{A}) = 0.6$

$P(B) = 0.3$

$P(A \cap B) = 0.1$

Find the value of the probability of the union, $P(A \cup B)$. Hint: calculate $P(A)$ first.

In a Leaving Certificate Mathematics exam, two events $A$ and $B$ from the same sample space are presented with probabilities $P(A) = 0.7$ and $P(B) = 0.5$. Cian writes in his working: "Since $P(A \cup B) = P(A) + P(B) = 0.7 + 0.5 = 1.2$, and by Kolmogorov's axioms no probability can exceed $1$, the problem statement contains an error and these two events cannot occur in the same experiment".\na) Identify and explain with reasoning the two conceptual errors made by Cian (one concerning the formula used and the other concerning the compatibility of the events).\nb) Using the properties derived from the axioms, calculate the minimum bound that $P(A \cap B)$ must take for the situation to be mathematically consistent, and determine the interval in which $P(A \cup B)$ can vary.

In the election of student representatives at a school, let $A$ be the event "the chosen person is in the Science track" with $P(A) = 0.65$, and let $B$ be the event "the chosen person participates in the debate club" with $P(B) = 0.45$.\na) Estimate, before performing any algebraic calculation, whether it is possible for both characteristics to be mutually exclusive ($A \cap B = \emptyset$). Justify your intuition based on the sum of their probabilities relative to the total sample space.\nb) Check your estimate by formally calculating the Fréchet bounds for the intersection; that is, the interval $[p_{\min}, p_{\max}]$ in which $P(A \cap B)$ must necessarily lie.\nc) Critically assess whether a prior estimate proposing $P(A \cap B) = 0.50$ would be admissible or if it contradicts the monotonicity property of probability ($A \cap B \subseteq B$).

In a study on digital content consumption in a Year 13 class, it is known that the probability of a student regularly using TikTok (event $A$) is $0.65$, the probability of using Instagram (event $B$) is $0.70$ and the probability of using neither of the two social networks is $0.15$.

Using the properties of probability, calculate:

1. a) The probability that a student, chosen at random, uses both social networks.
2. b) The probability that they use only one of the two.
3. c) Are the events $A$ and $B$ independent? Justify your answer.

Two esports teams, "Banshees" and "Shamrocks", participate in a tournament. The probability of the Banshees winning their match is $P(T) = 0.8$ and that of the Shamrocks winning theirs is $P(F) = 0.7$. Assuming that the results of both teams are independent, calculate the following probabilities using the algebra of events:

1. a) That both teams win their respective matches.
2. b) That at least one of the two wins.
3. c) That only the Banshees team wins.

Let $A$ and $B$ be two events of a random experiment such that we know $P(A) = 0.4$, $P(B) = 0.5$ and the probability of the union of their complements is $P(\bar{A} \cup \bar{B}) = 0.8$.

1. a) Calculate $P(A \cap B)$.
2. b) Find $P(A \cup B)$.
3. c) Determine whether the events $A$ and $B$ are compatible or mutually exclusive.
4. d) Calculate $P(A / \bar{B})$.

At a sports center, $60\%$ of the members do strength training (event $F$) and $45\%$ do yoga (event $Y$). It is also known that $20\%$ of those who do strength training are also enrolled in yoga.

Construct a contingency table or a Venn diagram to answer the following questions:

1. a) Probability that a randomly chosen member does neither strength training nor yoga.

1. b) If we know that a member does yoga, what is the probability that they do not do strength training?

1. c) Are the events "doing strength training" and "doing yoga" independent?

You can do it on paper and send a photo with your answer.

Let $A$ and $B$ be two events such that the probability that $B$ occurs given that $A$ has occurred is $P(B/A) = 0.25$. If we know that $P(A \cap B) = 0.15$ and that $P(B) = 0.45$:

1. a) Calculate $P(A)$.
2. b) Calculate $P(A \cup B)$.
3. c) Calculate the probability that $A$ does not occur given that $B$ has not occurred, that is, $P(\bar{A} / \bar{B})$.

A smart security system has two independent sensors, $S_1$ and $S_2$. The probability that sensor $S_1$ detects an intrusion is $0.92$ and the probability that sensor $S_2$ detects it is $0.85$. In the event that an intrusion occurs, calculate:

1. a) The probability that it is detected by both sensors simultaneously.
2. b) The probability that the intrusion is detected (that is, that at least one sensor works).
3. c) The probability that sensor $S_1$ fails but $S_2$ works correctly.

Given two events $A$ and $B$, it is known that $P(A) = 0.3$, $P(B) = 0.6$ and $P(A \cup B) = 0.72$.

1. a) Calculate $P(A \cap B)$.
2. b) Are the events $A$ and $B$ independent? Reason your answer by comparing $P(A \cap B)$ with the product of their individual probabilities.
3. c) Calculate $P(\bar{A} / B)$.

A streaming platform classifies its users as "Premium" ($P$) or "Standard" ($S$). It is estimated that $40\%$ are Premium. In addition, $70\%$ of Premium users watch science fiction series ($C$), whereas only $30\%$ of Standard users watch them. If a user is chosen at random:

1. a) Calculate the probability that they watch science fiction series.
2. b) If the chosen user is watching a science fiction series, what is the probability that they are a Premium user?
3. c) Are the events "Being a Premium user" and "Watching science fiction series" independent? Justify mathematically using properties of conditional probability.

Based on data adapted from the Central Statistics Office (CSO) on equipment and use of technologies among young people in Ireland aged 16 to 24, it is known that the probability that a young person uses a video-on-demand platform daily (event $V$) is $0.80$, and that they listen to podcasts regularly (event $P$) is $0.45$. Furthermore, the probability that they do neither of these two activities is only $0.05$.\na) Model the condition "does neither of the two activities" using set operations and, applying De Morgan's laws and the complement rule, calculate the probability of the union $P(V \cup P)$.\nb) Determine the probability that a randomly chosen young person does both activities, $P(V \cap P)$.\nc) An advertising agency claims that at least $20\,\%$ of young people listen to podcasts regularly but do not consume video on demand daily ($P \cap \overline{V}$). Using the partition of events into mutually exclusive parts ($P = (P \cap V) \cup (P \cap \overline{V})$), prove with reasoning whether this claim is true or false.

Consider two events $A$ and $B$ from a sample space whose probabilities depend on a real parameter $k$, such that $P(A) = 2k$ and $P(B) = 3k - 0.1$. It is known that the probability that exactly one of the two events occurs is equal to $0.5$.\na) Express the event "exactly one of the two events occurs" using set operations and show that its probability equals $P(A) + P(B) - 2 \times P(A \cap B)$.\nb) Assuming first that $A$ and $B$ are mutually exclusive events, determine the value of $k$ and rigorously verify that the resulting probabilities for $A$ and $B$ satisfy Kolmogorov's axioms ($0 \leq P(S) \leq 1$ and $P(A \cup B) \leq 1$).\nc) If it is now allowed that $A$ and $B$ can be compatible and $k = 0.20$ is set, determine the value that $P(A \cap B)$ must necessarily have to maintain the initial condition, and justify whether this value respects the theoretical bounds permitted for the intersection.

Let $A$, $B$, and $C$ be three events from the same random experiment such that they form a complete system of events. The following relationships between their probabilities are known:

• The probability of $A$ is twice that of $B$.

• The probability of the union of $B$ and $C$ is three times the probability of $B$.

Prove with reasoned arguments, using Kolmogorov's axioms and the properties of probability, whether it is possible that $P(C) = 0.4$. If not, determine the exact value of the probabilities of each event.

In a redundant security system for a server infrastructure, three control nodes ($N_1, N_2, N_3$) are installed. The system is known to fail if at least two of the nodes go down simultaneously. The failure probabilities of each node are $P(N_1) = 0.05$, $P(N_2) = 0.08$ and $P(N_3) = 0.10$.

Because they share the same power grid, the failures are not independent. It is known that:

• The probability that the first and second nodes fail is $P(N_1 \cap N_2) = 0.02$.

• The probability that the second and third nodes fail is $P(N_2 \cap N_3) = 0.03$.

• The probability that the first and third nodes fail is $P(N_1 \cap N_3) = 0.01$.

• The probability that all three fail at the same time is $P(N_1 \cap N_2 \cap N_3) = 0.005$.

1. a) Draw a Venn diagram with the three events $N_1$, $N_2$ and $N_3$ inside the sample space and calculate the probability of each of the eight regions (including the one outside the three circles).

1. b) Calculate the probability that the overall system does not fail, that is, that at most one node fails. Use the diagram from the previous part to justify your answer.

Formally prove the following generalisation of De Morgan's Laws for three events using exclusively the properties of set operations and the axioms of probability:

Then, use this result to solve the following problem: In a software development company, $60\%$ of the employees are proficient in Python, $40\%$ in Java and $30\%$ in C++. It is known that $20\%$ are proficient in Python and Java, $15\%$ in Java and C++, and $10\%$ in Python and C++. If only $5\%$ is proficient in all three languages, calculate the probability that a randomly chosen employee knows none of these three programming languages.

An aeronautical engineer analyses the reliability of a critical component. Let $A$ and $B$ be two events such that $P(A) = \frac{1}{3}$ and $P(B) = p$.

1. If $A$ and $B$ are independent, express the probability that exactly one of the two events occurs in terms of $p$.

2. If $A$ and $B$ are mutually exclusive, what is the maximum value that $p$ can take?

3. Determine the value of $p$ so that the probability that neither $A$ nor $B$ occurs is equal to $\frac{1}{4}$, assuming that $A$ and $B$ are independent.

In an IoT sensor network, two sensors $S_1$ and $S_2$ detect thermal anomalies independently. The probability that sensor $S_1$ detects an anomaly is $P(S_1) = x$ and that of sensor $S_2$ is $P(S_2) = y$.

A new event $Z$ is defined as the anomaly being detected by at least one of the two sensors. If we know that $P(Z) = 0.8$ and that the probability of both sensors failing simultaneously is four times the probability of both succeeding, set up a system of equations and calculate the values of $x$ and $y$ (assuming $x < y$).

Consider two events $A$ and $B$ of a random experiment. It is known that $P(A) = 0.6$ and $P(B) = 0.7$.

Based on Kolmogorov's axioms, prove that:

1. The minimum possible value for $P(A \cap B)$ is $0.3$.

2. The maximum possible value for $P(A \cap B)$ is $0.6$.

3. If a third event $C$ is added such that $P(C) = 0.8$, what is the minimum value of the probability that all three events occur simultaneously, $P(A \cap B \cap C)$? Justify your answer using Bonferroni's inequality: $P(A \cap B) \geq P(A) + P(B) - 1$.

In a biotechnology laboratory, the reaction of a protein to two reagents, $\alpha$ and $\beta$, is studied. The probability of reaction to $\alpha$ is $P(\alpha) = 0.4$. The probability of not reacting to either is $P(\alpha^c \cap \beta^c) = 0.3$.

If we denote by $R$ the event "the protein reacts to exactly one of the two reagents", and we know that $P(R) = 0.5$, determine:

1. a) The probability of reaction to reagent $\beta$.
2. b) Are the events of reacting to $\alpha$ and to $\beta$ independent? Justify your answer with calculations.
3. c) Are they compatible?

Let $E$ be a sample space and $A, B \subset E$ be two events with non-zero probabilities. The "symmetric difference" between the events is defined as $A \Delta B = (A \setminus B) \cup (B \setminus A)$.

If we know that $P(A \Delta B) = 0.6$ and that $P(A \cup B) = 0.8$:

1. Calculate $P(A \cap B)$.

2. If, in addition, $P(A | B) = 0.4$ holds, find the individual values of $P(A)$ and $P(B)$.

3. Using the values obtained, check whether the identity $P(A^c \cup B^c) = 1 - P(A \cap B)$ holds.

Prove, using exclusively Kolmogorov's axioms and the properties of operations with events (such as $A \cup B = A \cup (B \cap \overline{A})$, these last ones being mutually exclusive), the general formula for the probability of the union of any three events $A$, $B$ and $C$:

$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$

Let $A$ and $B$ be two events of a random experiment whose probabilities depend on a parameter $x$: $P(A) = \frac{x}{2}$, $P(B) = \frac{x}{3}$ and $P(A \cap B) = \frac{x}{6}$.

Knowing that the events $A$ and $B$ are not exhaustive (that is, $P(A \cup B) < 1$), determine the interval of possible values for the parameter $x$. For $x = 1.2$, calculate $P(\overline{A} \cup \overline{B})$.

In a space satellite, there are three sensors $S_1$, $S_2$ and $S_3$. From stress tests, the following failure probabilities are known:

• The probability that exclusively $S_1$ fails is $0.10$.

• The probability that exclusively $S_2$ fails is $0.08$.

• The probability that $S_1$ and $S_2$ fail simultaneously, but not $S_3$, is $0.05$.

• The probability that $S_3$ fails is $0.30$.

• The probability that all three fail simultaneously is $0.02$.

Knowing that the events "failure of $S_1$" and "failure of $S_3$" are mutually exclusive if $S_2$ does not fail, calculate the probability that at least one of the three sensors fails.

Two players, Ana and Blas, alternately toss a fair coin. The first player to obtain "Heads" wins. Ana tosses first. Define the infinite sample space of this experiment and, using the sum of a geometric progression, calculate the probability that Blas wins the game (that is, that the first "Heads" occurs on an even toss).

In the quality control of aeronautical components, three potential defects are analyzed: thermal ($T$), structural ($E$), and assembly ($M$). The following probability records are known:\n$P(T) = 0.12$, $P(E) = 0.08$, $P(M) = 0.06$;\n$P(T \cap E) = 0.03$, $P(T \cap M) = 0.02$, $P(E \cap M) = 0.01$;\nand the probability that all three defects occur simultaneously is $P(T \cap E \cap M) = 0.004$.\nIt is desired to calculate the probability that a randomly selected part is completely free of defects, that is, $P(\overline{T} \cap \overline{E} \cap \overline{M})$.\na) Compare two strategies to solve the problem:\n- Method 1: First calculate $P(T \cup E \cup M)$ using the principle of inclusion-exclusion for three events, and then apply the complement rule combined with De Morgan's laws.\n- Method 2: Partition the sample space into the eight disjoint regions of a Venn diagram, calculating one by one the probabilities of the seven defective regions to then subtract them from $1$.\nJustify which of the two methods is more efficient and less prone to algebraic errors in an assessment test.\nb) Apply the chosen strategy, justifying each equality with its corresponding axiomatic property to find the exact value of $P(\overline{T} \cap \overline{E} \cap \overline{M})$.\nc) Reason how the probability of obtaining a defect-free part would change if the three defects were pairwise mutually exclusive, and explain why in that case it would not be necessary to know any intersection.

Met Éireann, the Irish National Meteorological Service, monitors three severe weather phenomena in a mountain region during winter: snowfall warning ($N$), hurricane-force wind gust warning ($V$), and severe frost warning ($H$). Meteorological records establish that:\n- $P(N) = 0.40$, $P(V) = 0.35$, and $P(H) = 0.50$.\n- The dense cloud cover that accompanies snowfall prevents, for physical reasons, radiative frost at night from occurring, so the events $N$ and $H$ are mutually exclusive ($N \cap H = \emptyset$).\n- The probability of a joint warning for snow and wind is $P(N \cap V) = 0.15$.\na) Formulate the probability that at least one warning is activated, $P(N \cup V \cup H)$, using the general inclusion-exclusion rule for three events, and simplify it by deducing with reasoning the value of $P(N \cap V \cap H)$.\nb) If it is observed that $P(V \cap H) = 0.10$, calculate with justification the probability that the region spends a winter day completely free of meteorological warnings, and interpret the meaning of the result obtained.\nc) Suppose now that the specific value of $P(V \cap H)$ is unknown. Prove, by rigorously applying Kolmogorov's axioms ($P(E) = 1$ and additivity for disjoint events in subsets of $V$), that the value of $P(V \cap H)$ can only range within the interval $[0.10;\ 0.20]$ for the model to be mathematically consistent.