Total probability
A shop receives mobile phones from two different suppliers, and each one has its own failure rate. If you are asked "what is the probability that any phone from the shop is faulty?", you cannot look at just one supplier: you have to follow all the paths that lead to "faulty" and add them up. That is total probability, and you already know how to do it since you learned to multiply the branches of a tree — here you 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 merchandise and Factory 2 the remaining $60 \%$. It is known that $1 \%$…
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?
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: using $P(A)$ to find an unknown 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 ($R_1$), Centre ($R_2$) 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 ($R_1$), 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 ($R_2$).