Total Probability
A cell phone store gets its devices from two suppliers, and each one has its own failure rate. If someone asks you, "What is the probability that a randomly chosen phone from the store fails?", it is not enough to look at just one supplier: you have to follow all the paths that lead to "fails" and add them up. That is total probability. You already know how to multiply the branches of a tree; here we give it a name and a formula.
Complete system of events
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 action.
The law of total probability
A sportswear store receives its sneakers from two factories. Factory 1 supplies $40 \%$ of the stock and Factory 2 the remaining $60 \%$. It is known that $1 \%$ of the…
A sportswear store receives its sneakers from two factories. Factory 1 supplies $40 \%$ of the stock and Factory 2 the remaining $60 \%$. It is known that $1 \%$ of the sneakers from Factory 1 have a 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?
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…
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.
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 $M_3$…
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.
Three or more branches and complementary events
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…
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.
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…
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?
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…
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.
When a piece of data is missing: solving for a branch
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 $A$, $35\,\%$ 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 logistics company uses three routes to deliver packages in a large city: North ($R_1$), Center ($R_2$) and South ($R_3$). $30\,\%$ of deliveries go via $R_1$, $50\,\%$…
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 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 traveling on the Center route ($R_2$).
Back to the beginning: the probability that a randomly chosen phone from the store fails is the weighted sum of each supplier's failure rate, that is, the sum of all the branches of the tree that reach "fails."