Exercises: Total probability

On a shelf in a teenager's bedroom there are two boxes of video games. Box A contains 3 action games and 2 strategy games. Box B contains 2 action games and 4 strategy games. If we choose a box at random by flipping a coin and draw a game without looking, calculate the probability that the chosen video game is action.

A Year 13 student uses two apps to revise mathematics: App Alpha and App Beta. $60 \%$ of the time they use Alpha and the rest of the time they use Beta. The probability of finding a correctly solved exercise in Alpha is $0,9$, whereas in Beta it is $0,7$. If the student opens an app at random and selects an exercise, what is the probability that the exercise is correctly solved?

To get to school, a student can either go by bus or walk. On $70 \%$ of the days they take the bus and on $30 \%$ they go on foot. When going by bus, they are late $5 \%$ of the time, whereas if they walk, they are late on only $2 \%$ of the days. Calculate the probability that on any given day the student is late for class.

A sportswear store receives its sneakers from two factories. Factory 1 supplies $40 \%$ of the merchandise and Factory 2 the remaining $60 \%$. It is known that $1 \%$ of the sneakers from Factory 1 have some stitching defect, while for Factory 2 this percentage is $3 \%$. The tree diagram summarizes the data from the statement.

If a pair of sneakers is chosen at random from the warehouse, what is the probability that they are defective?

In a tennis club, $20 \%$ of the members are professionals and $80 \%$ are amateurs. In an internal tournament, the probability that a professional wins their first match is $0,95$, whereas for an amateur it is $0,40$. If we choose a member at random to watch their debut, what is the probability that they win their first match?

A content creator uploads videos to two platforms: YouTube and Instagram. $40 \%$ of their videos are published on YouTube and $60 \%$ on Instagram. The probability that a YouTube video exceeds 10 000 views is $0,8$, whereas on Instagram it is $0,3$. Choosing a video at random from their history, calculate the probability that it has more than 10 000 views.

At a recycling centre, waste arrives mixed in two lorries. Lorry A brings $55 \%$ of the waste and Lorry B brings $45 \%$. In Lorry A, $90 \%$ of the materials are recyclable, whereas in Lorry B only $75 \%$ are. If an item of waste is drawn at random from the total received, what is the probability that it is a recyclable material?

According to statistics from a café near a university campus, $50 \%$ of customers order coffee, $30 \%$ order tea, and $20 \%$ order juice. The probability that a customer who orders coffee wants it with milk is $0,8$; for tea it is $0,2$, and for juice it is $0$. If we select a customer at random, what is the probability that their drink contains milk?

In a garment manufacturing workshop, two sewing machines, $M_1$ and $M_2$, are used. Machine $M_1$ produces $70\,\%$ of the total output and has a defective garment rate of $2\,\%$, while machine $M_2$ produces the remaining $30\,\%$ with a defective garment rate of $5\,\%$. To calculate the probability that a randomly chosen garment is defective ($D$), Lucía wrote on her exam:

1. a) Explain with reasons why Lucía's approach is conceptually wrong and what mathematical absurdity would occur following her logic if both machines had, for example, a defective item rate of $60\,\%$.
2. b) Correct her solution by formally applying the law of total probability and calculate the actual probability that a garment chosen at random is defective.

In a large city, $80\,\%$ of daily public transport trips are made by metro and $20\,\%$ by bus. From past inspections by the transport authority, it is known that $1\,\%$ of metro users and $6\,\%$ of bus users travel without a valid ticket.

1. a) Before performing any exact calculations, make a reasoned estimate: indicate between which two percentage values the overall probability that a randomly chosen passenger travels without a ticket must necessarily lie, and explain why this probability will be significantly closer to $1\,\%$ than to $6\,\%$.
2. b) Apply the law of total probability to obtain the exact numerical value of this probability and formally verify whether it validates your previous estimate.

On a streaming platform, content is divided into three genres: Action (45%), Comedy (35%) and Documentaries (20%). It is known that the probability of a user watching a full film if it is Action is $0,7$; if it is Comedy it is $0,85$, and if it is a Documentary it is $0,5$. Choosing a film at random from the platform, calculate the probability that a user watches it to the end.

A secondary school offers three extracurricular activities: Robotics, Drama and Basketball. $25\ \%$ of the students are in Robotics, $30\ \%$ in Drama and the rest in Basketball. The percentage of students who pass all subjects is $80\ \%$ among those in Robotics, $90\ \%$ among those in Drama and $70\ \%$ among those in Basketball. If we select a student at random, what is the probability that they have passed all subjects?

A courier company uses three routes to deliver packages in a city. Route A carries $50\ \%$ of deliveries, route B $30\ \%$ and route C the remaining $20\ \%$. Due to traffic, the probability of a package experiencing a delay is $0,05$ on route A, $0,12$ on route B and $0,15$ on route C. Calculate the probability that a randomly chosen package arrives at its destination with a delay.

We have two urns with T-shirts from a streetwear brand. Urn 1 contains 6 white T-shirts and 4 black ones. Urn 2 contains 3 white T-shirts and 7 black ones. We roll a fair die: if a number less than 3 comes up, we draw a T-shirt from Urn 1; otherwise, we draw it from Urn 2. Find the probability that the drawn T-shirt is white.

In a tech shop, $60\ \%$ of the smartphones sold are brand X, $30\ \%$ are brand Y, and $10\ \%$ are brand Z. It is estimated that $2\ \%$ of brand X phones fail before the first year, whereas for brand Y the failure rate is $5\ \%$, and for brand Z it is $8\ \%$. Calculate the probability that a smartphone bought in that shop does not develop any faults during the first year.

A study on sleep habits in teenagers reveals that $40\ \%$ sleep less than 7 hours, $50\ \%$ sleep between 7 and 9 hours, and $10\ \%$ sleep more than 9 hours. The probability of having difficulty concentrating during the day is $0,8$ for those who sleep less than 7 hours, $0,2$ for the middle group and $0,1$ for those who sleep more than 9 hours. The diagram summarizes these data. Determine the probability that a randomly chosen teenager has difficulty concentrating.

Three machines, $M_1, M_2$ and $M_3$, are available to manufacture parts for electric scooters. $M_1$ produces $45\ \%$ of the parts, $M_2$ produces $30\ \%$, and $M_3$ produces $25\ \%$. The percentages of defective parts produced by each machine are, respectively, $3\ \%$, $4\ \%$, and $5\ \%$. If all the parts are mixed together in a warehouse, calculate the probability that a randomly chosen part is defective.

At a music festival, $70\ \%$ of the attendees are locals and $30\ \%$ are tourists. It is known that $20\ \%$ of the locals buy event merchandise, whereas among tourists this percentage rises to $60\ \%$. If we select a person at the festival at random, what is the probability that they do not buy merchandise?

At a driving school, students taking the practical driving test are divided into three groups according to their preparation: group $A$ (more than 30 practical lessons, $50\,\%$ of the total), group $B$ (between 15 and 30 lessons, $35\,\%$) and group $C$ (fewer than 15 lessons, the remainder). The probability of passing the test on the first attempt is $80\,\%$ for group $A$, $60\,\%$ for group $B$, and only $20\,\%$ for group $C$. We wish to find the probability that a randomly chosen student fails the test.

1. a) Propose and compare two strategies to calculate this probability:

• Strategy 1: Directly calculate the probability of failing by adding the products of the failure branches: $P(S) = \sum_{i} P(G_i)\cdot P(S|G_i)$.

• Strategy 2: Calculate the total probability of passing, $P(\text{Pass})$, and find the result using the complementary event: $P(S) = 1 - P(\text{Pass})$.

Justify which of the two strategies requires fewer arithmetic operations when the problem's data are given in terms of passes, and state what conditions the triplet $\{A, B, C\}$ must satisfy to constitute a complete system of events (a partition of the sample space).

1. b) Apply the chosen strategy and calculate the probability that a randomly selected student does not pass on the first attempt.

Based on reports from the National Statistics Institute (INE) regarding the profile of tourists staying overnight in a Mediterranean coastal region, it is determined that $50\,\%$ come from other regions of Spain (domestic tourism), $35\,\%$ come from other European Union countries, and the remaining $15\,\%$ come from outside the European Union (non-EU). Consumer studies indicate that the probability of a tourist spending more than $150$ euros per day is $0,20$ if they are domestic, $0,55$ if they come from the European Union, and $0,80$ if they are non-EU.

1. a) Define the events involved precisely, justify why the origins form a complete system of events, and model the situation by applying the law of total probability to determine the probability that a randomly chosen visitor spends more than $150$ euros per day.
2. b) The municipality's commercial revitalisation plan has set a goal that at least $40\,\%$ of tourists exceed this daily spending threshold of $150$ euros. Conclude with justification whether the current distribution of origins achieves this goal, and explain the impact on the overall spending probability if $10\,\%$ of total visitors shift from domestic to non-EU tourism.

In a biotechnology laboratory, three different bacterial strains are being studied: $A$, $B$ and $C$. It is known that $40\,\%$ of the samples contain strain $A$, $35\,\%$ strain $B$ and the rest strain $C$. The probability of a specific antibiotic eliminating the bacterium is $0,8$ for strain $A$ and $0,5$ for strain $B$. If it is known that the total probability of the antibiotic eliminating a randomly chosen bacterium is $62\,\%$, calculate with reasoning the probability that the antibiotic is effective against strain $C$.

A social media algorithm classifies videos into three categories according to their content: "Gaming" ($G$), "Music" ($M$) and "Comedy" ($H$). $50\,\%$ of the videos are Gaming, $30\,\%$ Music and $20\,\%$ Comedy. The probability of a user stopping scrolling and watching a full video is $0,3$ if it is Gaming and $0,6$ if it is Music. It has been observed that Comedy videos are particularly viral, doubling the probability of being watched in full compared to Gaming videos. Calculate the probability that a randomly chosen user watches a full video, regardless of its category.

We have two urns with different compositions. Urn 1 contains 4 red balls and 6 blue balls. Urn 2 contains 7 red balls and 3 blue balls. A fair die is rolled: if a prime number is rolled, a ball is drawn from Urn 1 and placed into Urn 2, and then a final ball is drawn from this second urn. If a non-prime number is rolled, a ball is drawn from Urn 2 and placed into Urn 1, and then the final ball is drawn from Urn 1. Calculate the probability that the ball drawn at the end of the whole process is red.

In a smartphone components factory, $5\,\%$ of the screens produced in the morning are defective, whereas in the afternoon the defect percentage rises to $8\,\%$. The production volume is constant throughout the day, but due to a fault in the storage system, the batches from three consecutive days are mixed. On the third day, a new machine improved performance, making only $2\,\%$ of that day's screens defective. If a screen is chosen at random from the warehouse (which contains the total production of the three days), calculate the probability that the screen is not defective.

A 2nd year Bachillerato student sits a multiple choice exam. For each question, the student either knows the answer or does not know it and chooses one at random from the 4 possible options. It is estimated that the student knows the answer to $70\,\%$ of the syllabus questions. However, the exam is poorly designed and $10\,\%$ of the questions that the student "knows" have ambiguous wording that makes them fail despite knowing the concept. Calculate the probability that the student answers a randomly chosen question correctly.

A logistics company uses three routes to deliver packages in a large city: North ($R_1$), Centre ($R_2$) and South ($R_3$). $30\,\%$ of deliveries go via $R_1$, $50\,\%$ via $R_2$ and $20\,\%$ via $R_3$. Delays are common due to traffic: on $R_1$ there is a delay $15\,\%$ of the time and on $R_3$ $25\,\%$. It is known that the probability of a package arriving on time is $0,82$. Determine the percentage of packages that experience delays when travelling on the Centre route ($R_2$).

In an e-sports tournament, a professional player has to face three difficulty levels: Easy, Normal and Legend. At the start, a level is assigned to them at random by a draw where the Easy level has probability $p$, Normal $2p$ and Legend $0,1$. If the player completes the assigned level, they win the tournament. The probabilities of winning are $0,9$ on Easy, $0,6$ on Normal and $0,2$ on Legend.

1. a) Find the value of $p$ so that the level distribution is a valid probability distribution.
2. b) Calculate the probability that the player wins the tournament.

A Spanish 40-card deck and a six-sided die are available. A card is drawn from the deck. If it is a face card, the die is rolled once. If the card is not a face card, the die is rolled twice and the scores are added together.

1. a) Calculate the probability that the sum of the scores obtained is equal to 6.
2. b) What is the probability that at least one 5 is rolled in total across the rolls made (whether one or two rolls)?

The risk control department of a bank classifies mortgage applications into three creditworthiness profiles: Favourable ($N_1$), Intermediate ($N_2$), and Unfavourable ($N_3$). In its current portfolio, $55\,\%$ of applications correspond to $N_1$ and $30\,\%$ to $N_2$. From historical data, it is estimated that the default probability is $0,01$ for $N_1$ and $0,05$ for $N_2$.

1. a) The banking supervisor requires that the overall default probability of the portfolio does not exceed $3\,\%$. Determine algebraically the maximum value that the conditional default probability of the Unfavourable group, $P(\text{Default}|N_3)$, can take so that the bank complies with the regulations without having to reject or alter the current proportions of clients in each group.
2. b) Suppose that, following a general economic downturn, the default probability of the Unfavourable profile suddenly rises to $0,12$. To maintain the overall default rate at exactly $3\,\%$, the bank decides to restructure the portfolio by modifying the share of accepted applications from $N_1$ and $N_3$, while keeping the share of the Intermediate group $N_2$ unchanged at $30\,\%$. Formulate the equation of the law of total probability in terms of the new proportion $p$ of type $N_1$ clients and solve in a reasoned manner for the minimum percentage of favourable applications that the institution must attract.

An olive oil mill in Jaén classifies the olives entering its facilities into two categories: Early Harvest ($T$) and Traditional Harvest ($R$). A proportion $p$ of the olives is harvested early ($0 < p < 1$) and the rest through traditional harvesting. Due to the characteristics of the fruit, the probability of a batch achieving the highest Extra Virgin Olive Oil (EVOO) rating is $0,90$ if it comes from an early harvest and $0,55$ if it comes from a traditional harvest.

1. a) Express the total probability of any batch obtaining the EVOO rating as a linear function of $p$, $f(p)$. Interpret the agronomic and mathematical meaning of the slope and the y-intercept of this line in the context of the problem.
2. b) For the upcoming season, the technical management is considering two improvement plans with identical budget costs:

• Plan 1: Maintain the current proportion of early harvest at $p = 0,20$, but incorporate a cooling system during malaxation that raises the probability of obtaining EVOO from the traditional harvest from $0,55$ to $0,70$.

• Plan 2: Keep the current machinery (leaving the efficiencies at $0,90$ and $0,55$), but provide financial incentives to farmers to harvest earlier, increasing the proportion of early harvest from $p = 0,20$ to $p = 0,60$.

Calculate using the law of total probability the final proportion of EVOO that would be achieved with each plan, compare them by justifying which is more advantageous to maximise the production of top-quality oil, and reflect on what operational risk the cooperative would take on if it chooses Plan 2 over Plan 1.