Exercises: Trees

A box contains $4$ red balls and $6$ blue balls. A ball is drawn at random, its color is noted, it is returned to the box and a second ball is drawn. Draw the corresponding tree diagram and calculate the probability of getting two blue balls.

A café offers two types of menu: $60\,\%$ of the customers choose the daily menu and $40\,\%$ choose the vegetarian menu. Of those who choose the daily menu, $70\,\%$ order coffee for dessert. Of those who choose the vegetarian menu, $80\,\%$ order tea. Calculate the probability that a randomly chosen customer has had the daily menu and has also ordered coffee.

Complete the following tree diagram that represents the flipping of a biased coin twice consecutively, knowing that the probability of getting heads is $\frac{1}{3}$. From the tree, calculate the probability of getting heads first and then tails.

In a wardrobe there are $5$ white t-shirts and $3$ black ones. Two t-shirts are chosen at random, one after the other and without returning the first one to the wardrobe. Calculate the probability that both t-shirts are white.

We roll a six-sided die and, then, we draw a card from a deck of $40$ cards. If an even number is rolled on the die, we win if we draw a King. If an odd number is rolled on the die, we win if we draw an Ace. Calculate the probability of rolling an even number on the die and also winning.

To go to work, Emma can go by bus (with probability $0,7$) or by bicycle (with probability $0,3$). If she goes by bus, the probability of arriving late is $0,2$. If she goes by bicycle, the probability of arriving late is $0,05$. What is the probability that Emma goes by bicycle and arrives on time?

In an urn there are $2$ green balls and $3$ yellow balls. We draw a ball, look at it and leave it out. Then we draw another ball. What is the probability that the first is green and the second is yellow?

A basketball player has a probability of $0,8$ of scoring a free throw. If they shoot two free throws independently, draw the probability tree and determine what is the probability that they miss both shots.

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.

In a tennis match, if a player's first serve goes into the box (probability $0,6$), they win the point with probability $0,7$. If the first serve misses, they hit a second serve that goes in with probability $0,8$. If the second serve goes in, they win the point with probability $0,5$. If the second serve also misses, they double fault and lose the point automatically. Calculate the total probability that the player wins the point on that service.

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 lands on heads, we draw a ball from urn A; if it lands on tails, we draw a ball from urn B. What is the probability that the drawn ball is red?

A student has to take two exams on the same day. The probability of passing the first one is $0,6$. If they pass the first one, they get motivated and the probability of passing the second one rises to $0,8$. However, if they fail the first one, they get discouraged and the probability of passing the second one drops to $0,3$. Draw the tree diagram and calculate the probability that they pass at least one of the two exams.

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.

A bag contains $5$ white balls and $4$ black balls. Balls are drawn one by one and without replacement until a black ball is drawn. What is the probability of having to make exactly three draws for the game to end?

The weather in a coastal city follows a curious pattern: if a day is sunny, the probability that it rains the next day is $0,2$. If it rains one day, the probability that it continues raining the next day is $0,6$. Knowing that today is Monday and it is sunny, what is the probability that it rains on Wednesday? Draw the corresponding tree diagram for Tuesday and Wednesday.

We have a deck of $40$ cards. Three cards are drawn consecutively without replacement. Calculate the probability that all three cards are of the same suit (coins, cups, swords or clubs).

There are two boxes. Box 1 contains $4$ blue counters and $2$ red counters. Box 2 contains $3$ blue counters and $5$ red counters. A counter is drawn at random from box 1 and placed into box 2. Then, a counter is drawn from box 2. Calculate the probability that the counter drawn from box 2 is blue.

In an urn there are $x$ white balls and $4$ black balls. Two balls are drawn without replacement. Knowing that the probability of both balls being white is $\frac{1}{6}$, set up the corresponding equation using a tree diagram and determine the initial number of white balls $x$.

Two e-sports teams, Alpha and Beta, play the final of a tournament in a best-of-three match (the first to win two games is the champion). Alpha is the better team, so their probability of winning any game is $0,6$. Games cannot end in a draw. Draw the tree diagram of the possible outcomes of the final and calculate the probability that team Beta wins the tournament.

An alarm system in a vault has 3 motion sensors. The probability that sensor A fails during an intrusion is $0,1$; the probability that sensor B fails is $0,05$; and the probability that sensor C fails is $0,2$. The sensors act completely independently. Draw the corresponding tree diagram and calculate the probability that, during a real intrusion, at least one sensor is activated and sounds the alarm.

In a city, $20\,\%$ of drivers are young (under 25 years old) and $80\,\%$ are adults. The probability of a young driver having an accident in a year is $0,15$, while for an adult it is $0,05$. A driver is selected at random and it turns out they 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).

In a board game, a player must cross a pit by jumping over three successive platforms. The probability of slipping on the first platform is $\frac{1}{10}$. If they pass the first one, the probability of slipping on the second is $\frac{1}{5}$. If they pass the second, the probability of slipping on the third is $\frac{1}{4}$. If they slip on any platform, they fall into the pit and lose. What is the probability that the player crosses the pit successfully? If we know that the player fell into the pit, what is the probability that they fell exactly on the second platform?

An urn contains $5$ balls numbered from $1$ to $5$. Balls are drawn one by one, without replacement, until the sum of the drawn numbers is greater than or equal to $6$. Draw the tree diagram with all possible branches knowing that the ball with the number $2$ was drawn in the first draw. From your tree, calculate the probability that the game ends exactly on the third draw.

In a competition, an archer has a maximum of three arrows to hit the target. The probability of hitting with the first arrow is $0,4$. If they miss, they adjust their sight and the probability of hitting with the second is $0,6$. If they miss again, they get nervous and the probability of hitting with the third is $0,3$. The archer stops shooting the moment they hit the target. Calculate the probability that the archer needs to take more than one shot knowing that they finally managed to hit the target.