Tree Diagrams
You already know how to multiply branches and add paths. What is missing is a way of organising it all without getting lost when the experiment has three, four or more stages: the tree diagram. It is the tool you will use systematically in Bachillerato for total probability and Bayes' theorem, so it is worth mastering how to construct one rigorously, not just intuitively.
Constructing a tree diagram
Complete the following tree diagram representing the tossing of a biased coin twice consecutively, given that the probability of getting heads is $\frac{1}{3}$. From the…
Complete the following tree diagram representing the tossing of a biased coin twice consecutively, given that the probability of getting heads is $\frac{1}{3}$. From the tree diagram, calculate the probability of getting heads first and then tails.
A basketball player has a probability of $0,8$ of scoring a free throw. If they take two free throws independently, draw the probability tree diagram and determine the…
A basketball player has a probability of $0,8$ of scoring a free throw. If they take two free throws independently, draw the probability tree diagram and determine the probability that they miss both shots.
You can do it on paper and send a photo with your answer.
Paths and outcomes: multiply and add
In a wardrobe there are $5$ white T-shirts and $3$ black T-shirts. Two T-shirts are chosen at random, one after the other and without replacing the first back into the…
In a wardrobe there are $5$ white T-shirts and $3$ black T-shirts. Two T-shirts are chosen at random, one after the other and without replacing the first back into the wardrobe. Calculate the probability that both T-shirts are white.
A Spanish pack of $40$ cards is available. Three cards are drawn consecutively without replacement. Calculate the probability that all three cards are of the same suit…
A Spanish pack of $40$ cards is available. Three cards are drawn consecutively without replacement. Calculate the probability that all three cards are of the same suit (coins, cups, swords or clubs).
Total probability read from the tree
In an electronic components factory, three machines (A, B, and C) produce $50\,\%$, $30\,\%$, and $20\,\%$ of the parts, respectively. The percentages of defective parts…
In an electronic components factory, three machines (A, B, and C) produce $50\,\%$, $30\,\%$, and $20\,\%$ of the parts, respectively. The percentages of defective parts produced by each machine are $2\,\%$, $4\,\%$, and $5\,\%$. Use the tree diagram to calculate the probability that a randomly chosen part is defective.
We have two urns. Urn A contains $3$ red balls and $2$ black balls. Urn B contains $4$ red balls and $5$ black balls. We toss a coin: if it comes up heads, we draw a…
We have two urns. Urn A contains $3$ red balls and $2$ black balls. Urn B contains $4$ red balls and $5$ black balls. We toss a coin: if it comes up heads, we draw a ball from urn A; if it comes up tails, we draw a ball from urn B. What is the probability that the drawn ball is red?
In a diagnostic test for an allergy, $5\,\%$ of the population actually has the allergy. If a person has the allergy, the test is positive $90\,\%$ of the time. If they…
In a diagnostic test for an allergy, $5\,\%$ of the population actually has the allergy. If a person has the allergy, the test is positive $90\,\%$ of the time. If they do not have it, the test is negative $95\,\%$ of the time. Draw a tree diagram to illustrate the situation and calculate the probability that a randomly chosen person gets a positive result on the test.
You can do it on paper and send a photo with your answer.
Bayes' theorem from the tree
The tree can also answer the reverse question: given the final outcome ($B$ has occurred), which branch of the first stage is most likely to have led there?
In a city, $20\,\%$ of drivers are young (under 25 years old) and $80\,\%$ are adults. The probability that a young driver has an accident in a year is $0,15$, whereas…
In a city, $20\,\%$ of drivers are young (under 25 years old) and $80\,\%$ are adults. The probability that a young driver has an accident in a year is $0,15$, whereas for an adult it is $0,05$. A driver is selected at random and turns out to have had an accident this year. Using the branches of a tree diagram, calculate the probability that this driver is young. (Express the result as an irreducible fraction).
A vault alarm system has 3 motion sensors. The probability that sensor A fails during an intrusion is $0,1$; that sensor B fails is $0,05$; and that sensor C fails is…
A vault alarm system has 3 motion sensors. The probability that sensor A fails during an intrusion is $0,1$; that sensor B fails is $0,05$; and that sensor C fails is $0,2$. The sensors act completely independently. Draw the corresponding tree diagram and calculate the probability that, during an actual intrusion, at least one sensor activates and sounds the alarm.
You can do it on paper and send a photo with your answer.