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 an action game.

A high school senior uses two apps to review math: App Alpha and App Beta. They use Alpha $60 \%$ of the time and Beta the rest of the time. 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 go to school, a student can go by bus or walk. They use the bus $70 \%$ of the days and walk $30 \%$ of the days. When they go by bus, they arrive late $5 \%$ of the time, whereas if they walk, they arrive late only $2 \%$ of the days. Calculate the probability that on any given day the student arrives late for class.

A sporting goods store receives its sneakers from two factories. Factory 1 supplies $40 \%$ of the goods and Factory 2 the remaining $60 \%$. It is known that $1 \%$ of the sneakers from Factory 1 have some stitching defect, whereas in Factory 2 this percentage is $3 \%$. 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 of a professional winning 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 of a YouTube video exceeding 10,000 views is $0,8$, whereas on Instagram it is $0,3$. When choosing a video at random from their history, calculate the probability that it has more than 10,000 views.

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

According to the statistics of a coffee shop 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?

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

A high 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 their classes 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 their classes?

A courier company uses three routes to deliver packages in a city. $50\ \%$ of the deliveries travel via route A, $30\ \%$ via route B, and the remaining $20\ \%$ via route C. Due to traffic, the probability of a package suffering 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 late at its destination.

We have two urns with T-shirts from a streetwear brand. Urn 1 contains 6 white and 4 black T-shirts. Urn 2 contains 3 white and 7 black T-shirts. We roll a fair die: if a number less than 3 is rolled, 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 an electronics store, $60\ \%$ 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 store does not present any faults during the first year.

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

There are three machines, $M_1, M_2$ and $M_3$, which 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 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 from the festival at random, what is the probability that they do not buy merchandise?

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 $A$, $35\,\%$ contain strain $B$, and the rest contain strain $C$. The probability that a specific antibiotic eliminates the bacteria is $0,8$ for strain $A$ and $0,5$ for strain $B$. If it is known that the total probability that the antibiotic eliminates a bacterium chosen at random 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 "Humor" ($H$). $50\,\%$ of the videos are Gaming, $30\,\%$ are Music, and $20\,\%$ are Humor. The probability that a user stops scrolling and watches a full video is $0,3$ if it is Gaming and $0,6$ if it is Music. It has been observed that Humor videos are especially viral, doubling the probability of being watched in full compared to Gaming videos. Calculate the probability that a user chosen at random 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 component factory, $5\,\%$ of the screens produced in the morning are defective, whereas in the afternoon the percentage of defects rises to $8\,\%$. The production volume is constant throughout the day, but due to a failure in the storage system, the batches from three consecutive days are mixed. On the third day, a new machine improved performance, meaning that only $2\,\%$ of the screens from that day were defective. If a screen is chosen at random from the warehouse (which contains the total production from the three days), calculate the probability that the screen is not defective.

A high school senior takes 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 questions on the syllabus. However, the exam is poorly designed and $10\,\%$ of the questions that the student "knows" have ambiguous wording that causes them to get it wrong 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 ($R_1$), Center ($R_2$), and South ($R_3$). $30\,\%$ of the 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 $R_3$ $25\,\%$ of the time. It is known that the probability of a package arriving on time is $0,82$. Determine the percentage of packages that suffer delays when traveling on the Center 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 at random by a draw where the Easy level has a probability of $p$, Normal $2p$, and Legend $0,1$. If the player beats 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 distribution of levels is a valid probability distribution.
2. b) Calculate the probability that the player wins the tournament.

We have a 40-card Spanish deck and a six-sided die. 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.

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