The Law of Total Probability
A phone store at the mall buys its phones from two suppliers, and each supplier has its own failure rate. Someone asks, "What's the probability that a random phone from this store fails in the first year?" You can't look at just one supplier. You have to follow every path that ends in "fails" and add them up. That is the Law of Total Probability, and you already know the idea from tree diagrams. Here you give it a name and a formula.
Partitions of the sample space
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…
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…
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?
The Law of Total Probability
That answers the question from the top of the page: a random phone fails with probability $0.0575$, or $5.75\%$. It sits between the two supplier rates, closer to $X$'s because $X$ supplies more phones.
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…
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.
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…
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?
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…
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.
Three or more branches, and complements
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…
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.
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…
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.
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…
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.
When a piece is missing
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 $A$,…
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 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 $R_1$,…
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…
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.
- a) Find the value of $p$ so that the distribution of levels is a valid probability distribution.
- b) Calculate the probability that the player wins the tournament.
Multi-stage setups
Harder problems still follow the same plan: first choose the partition (what happens first), then work out the conditional probability of the final event in each case. Often that conditional probability needs its own small calculation, because the contents of an urn changed or the number of rolls depends on the first stage.
Try drawing from the bag a few times. When a ball is moved from one bag to another before the final draw, the composition of the second bag depends on what was moved. You must condition on that too.
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…
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.
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…
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.
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…
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.
- a) Calculate the probability that the sum of the scores obtained is equal to 6.
- b) What is the probability of obtaining at least a 5 in the total number of rolls made (whether one or two rolls)?