Tree Diagrams

1. A company assembles drills using components from two sources. Goodbuy supplies \(85 \%\) of the components and Amart supplies the rest. It is known that \(3 \%\) of the components supplied by Goodbuy are faulty and \(6 \%\) of those supplied by Amart are faulty.
   1. Represent this information on a tree diagram.
   An assembled drill is selected at random.
2. Find the probability that it is not faulty.

1. A company assembles drills using components from two sources. Goodbuy supplies \(85 \%\) of the components and Amart supplies the rest. It is known that \(3 \%\) of the components supplied by Goodbuy are faulty and \(6 \%\) of those supplied by Amart are faulty.
   1. Represent this information on a tree diagram.
   An assembled drill is selected at random.
2. Find the probability that it is not faulty.

4 It is proposed to introduce, for all males at age 60, screening tests, A and B, for a certain disease. Test B is administered only when the result of Test A is inconclusive. It is known that 10\% of 60-year-old men suffer from the disease. For those 60 -year-old men suffering from the disease:

• Test A is known to give a positive result, indicating a presence of the disease, in \(90 \%\) of cases, a negative result in \(2 \%\) of cases and a requirement for the administration of Test B in \(8 \%\) of cases;
• Test B is known to give a positive result in \(98 \%\) of cases and a negative result in 2\% of cases.

For those 60 -year-old men not suffering from the disease:

• Test A is known to give a positive result in \(1 \%\) of cases, a negative result in \(80 \%\) of cases and a requirement for the administration of Test B in 19\% of cases;
• Test B is known to give a positive result in \(1 \%\) of cases and a negative result in \(99 \%\) of cases.

7 A bag contains three 1 p coins and seven 2 p coins. Coins are removed at random one at a time, without replacement, until the total value of the coins removed is at least 3p. Then no more coins are removed.

1. Copy and complete the probability tree diagram. First coin \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-4_350_317_1279_568} Find the probability that
2. exactly two coins are removed,
3. the total value of the coins removed is 4p.

4 To gain a place at a science college, students first have to pass a written test and then a practical test.
Each student is allowed a maximum of two attempts at the written test. A student is only allowed a second attempt if they fail the first attempt. No student is allowed more than one attempt at the practical test. If a student fails both attempts at the written test, then they cannot attempt the practical test. The probability that a student will pass the written test at the first attempt is 0.8 . If a student fails the first attempt at the written test, the probability that they will pass at the second attempt is 0.6 . The probability that a student will pass the practical test is always 0.3 .

1. Draw a tree diagram to represent this information, showing the probabilities on the branches.
2. Find the probability that a randomly chosen student will succeed in gaining a place at the college.
   [0pt] [2]
3. Find the probability that a randomly chosen student passes the written test at the first attempt given that the student succeeds in gaining a place at the college.

3 Candidates applying for jobs in a large company take an aptitude test, as a result of which they are either accepted, rejected or retested, with probabilities \(0.2,0.5\) and 0.3 respectively. When a candidate is retested for the first time, the three possible outcomes and their probabilities remain the same as for the original test. When a candidate is retested for the second time there are just two possible outcomes, accepted or rejected, with probabilities 0.4 and 0.6 respectively.

1. Draw a probability tree diagram to illustrate the outcomes.
2. Find the probability that a randomly selected candidate is accepted.
3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted.

A jar contains 2 red, 1 blue and 1 green bead. Two beads are drawn at random from the jar without replacement.

1. In the space below, draw a tree diagram to illustrate all the possible outcomes and associated probabilities. State your probabilities clearly. [3]
2. Find the probability that a blue bead and a green bead are drawn from the jar. [2]

2 The probability that Henk goes swimming on any day is 0.2 . On a day when he goes swimming, the probability that Henk has burgers for supper is 0.75 . On a day when he does not go swimming the probability that he has burgers for supper is \(x\). This information is shown on the following tree diagram. \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-2_693_1038_845_555} The probability that Henk has burgers for supper on any day is 0.5 .

1. Find \(x\).
2. Given that Henk has burgers for supper, find the probability that he went swimming that day.

Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_5} A day is selected at random. Find the probability that

1. the weather is wet and Andy travels by bus, [2]
2. Andy walks or travels by bike, [3]
3. the weather is dry given that Andy walks or travels by bike. [3]

5 Maryam has 7 sweets in a tin; 6 are toffees and 1 is a chocolate. She chooses one sweet at random and takes it out. Her friend adds 3 chocolates to the tin. Then Maryam takes another sweet at random out of the tin.

1. Draw a fully labelled tree diagram to illustrate this situation.
2. Draw up the probability distribution table for the number of toffees taken.
3. Find the mean number of toffees taken.
4. Find the probability that the first sweet taken is a chocolate, given that the second sweet taken is a toffee.

1. There are 7 red counters, 3 blue counters and 2 yellow counters in a bag. Gina selects a counter at random from the bag and keeps it. If the counter is yellow she does not select any more counters. If the counter is not yellow she randomly selects a second counter from the bag.
   1. Complete the tree diagram.
   First Counter
   Second Counter \includegraphics[max width=\textwidth, alt={}, center]{a439724e-b570-434d-bf75-de2b50915042-02_1147_1081_603_397} Given that Gina has selected a yellow counter,
2. find the probability that she has 2 counters.

1. In a sixth form college each student in Year 12 and Year 13 is either left-handed (L) or right-handed (R).

The partially completed tree diagram, where \(p\) is a probability, gives information about these students. \includegraphics[max width=\textwidth, alt={}, center]{86446ce3-496a-4f02-9566-9b207bac9efa-10_960_981_477_543}

1. Complete the tree diagram, in terms of \(p\) where necessary. The probability that a student is left-handed is 0.11
2. Find the value of \(p\)
3. Find the probability that a student selected at random is in Year 12 and left-handed. Given that a student is right-handed,
4. find the probability that the student is in Year 12

3 Jimmy and Alan are playing a tennis match against each other. The winner of the match is the first player to win three sets. Jimmy won the first set and Alan won the second set. For each of the remaining sets, the probability that Jimmy wins a set is

• 0.7 if he won the previous set,
• 0.4 if Alan won the previous set.

It is not possible to draw a set.

1. Draw a probability tree diagram to illustrate the possible outcomes for each of the remaining sets.
2. Find the probability that Alan wins the match.
3. Find the probability that the match ends after exactly four sets have been played.

1. Bag A contains 7 blue discs, 4 red discs and 1 yellow disc. Two discs are drawn at random from bag A without replacement. Find the probability that exactly one of the discs is blue. [2 marks]
2. Bag A contains 7 blue discs, 4 red discs and 1 yellow disc. Bag B contains 3 blue discs and 6 red discs. A disc is drawn at random from Bag A and placed in Bag B. A disc is then drawn at random from Bag B. Find the probability that the disc drawn from Bag B is red. [3 marks]

8 A game is played with a fair, six-sided die which has 4 red faces and 2 blue faces. One turn consists of throwing the die repeatedly until a blue face is on top or until the die has been thrown 4 times.

1. In the answer book, complete the probability tree diagram for one turn. \includegraphics[max width=\textwidth, alt={}, center]{e5957185-5fe3-45d9-9ab3-c2aab9cbd8dd-5_314_302_1000_884}
2. Find the probability that in one particular turn the die is thrown 4 times.
3. Adnan and Beryl each have one turn. Find the probability that Adnan throws the die more times than Beryl.
4. State one change that needs to be made to the rules so that the number of throws in one turn will have a geometric distribution.

2 A bag contains 5 black discs and 3 red discs. A disc is selected at random from the bag. If it is red it is replaced in the bag. If it is black, it is not replaced. A second disc is now selected at random from the bag. Find the probability that

1. the second disc is black, given that the first disc was black,
2. the second disc is black,
3. the two discs are of different colours.

1. Two bags, \(\mathbf { A }\) and \(\mathbf { B }\), each contain balls which are either red or yellow or green.

Bag A contains 4 red, 3 yellow and \(n\) green balls.
Bag \(\mathbf { B }\) contains 5 red, 3 yellow and 1 green ball.
A ball is selected at random from bag \(\mathbf { A }\) and placed into bag \(\mathbf { B }\).
A ball is then selected at random from bag \(\mathbf { B }\) and placed into bag \(\mathbf { A }\).
The probability that bag \(\mathbf { A }\) now contains an equal number of red, yellow and green balls is \(p\). Given that \(p > 0\), find the possible values of \(n\) and \(p\).

3 One train leaves a station each hour. The train is either on time or late. If the train is on time, the probability that the next train is on time is 0.95 . If the train is late, the probability that the next train is on time is 0.6 . On a particular day, the 0900 train is on time.

1. Illustrate the possible outcomes for the 1000,1100 and 1200 trains on a probability tree diagram.
2. Find the probability that
   (A) all three of these trains are on time,
   (B) just one of these three trains is on time,
   (C) the 1200 train is on time.
3. Given that the 1200 train is on time, find the probability that the 1000 train is also on time.

7 Box \(A\) contains 5 red paper clips and 1 white paper clip. Box \(B\) contains 7 red paper clips and 2 white paper clips. One paper clip is taken at random from box \(A\) and transferred to box \(B\). One paper clip is then taken at random from box \(B\).

1. Find the probability of taking both a white paper clip from box \(A\) and a red paper clip from box \(B\).
2. Find the probability that the paper clip taken from box \(B\) is red.
3. Find the probability that the paper clip taken from box \(A\) was red, given that the paper clip taken from box \(B\) is red.
4. The random variable \(X\) denotes the number of times that a red paper clip is taken. Draw up a table to show the probability distribution of \(X\).

8 A game uses an unbiased die with faces numbered 1 to 6 . The die is thrown once. If it shows 4 or 5 or 6 then this number is the final score. If it shows 1 or 2 or 3 then the die is thrown again and the final score is the sum of the numbers shown on the two throws.

1. Find the probability that the final score is 4 .
2. Given that the die is thrown only once, find the probability that the final score is 4 .
3. Given that the die is thrown twice, find the probability that the final score is 4 .

7 One train leaves a station each hour. The train is either on time or late. If the train is on time, the probability that the next train is on time is 0.95 . If the train is late, the probability that the next train is on time is 0.6 . On a particular day, the 0900 train is on time.

1. Illustrate the possible outcomes for the 1000,1100 and 1200 trains on a probability tree diagram.
2. Find the probability that
   (A) all three of these trains are on time,
   (B) just one of these three trains is on time,
   (C) the 1200 train is on time.
3. Given that the 1200 train is on time, find the probability that the 1000 train is also on time. 3
4. Write any calculations on page 5. \includegraphics[max width=\textwidth, alt={}, center]{091d6f43-ad01-4849-9f3c-3e58349aa169-4_2276_1490_324_363}

1 Laura frequently flies to business meetings and often finds that her flights are delayed. A flight may be delayed due to technical problems, weather problems or congestion problems, with probabilities \(0.2,0.15\) and 0.1 respectively. The tree diagram shows this information. \includegraphics[max width=\textwidth, alt={}, center]{10679ff3-494d-4f4e-a38a-0832faa91690-1_605_1650_534_284}

1. Write down the values of the probabilities \(a , b\) and \(c\) shown in the tree diagram. One of Laura's flights is selected at random.
2. Find the probability that Laura's flight is not delayed and hence write down the probability that it is delayed.
3. Find the probability that Laura's flight is delayed due to just one of the three problems.
4. Given that Laura's flight is delayed, find the probability that the delay is due to just one of the three problems.
5. Given that Laura's flight has no technical problems, find the probability that it is delayed.
6. In a particular year, Laura has 110 flights. Find the expected number of flights that are delayed.

2 A sports event is taking place for 4 days, beginning on Sunday. The probability that it will rain on Sunday is 0.4 . On any subsequent day, the probability that it will rain is 0.7 if it rained on the previous day and 0.2 if it did not rain on the previous day.

1. Find the probability that it does not rain on any of the 4 days of the event.
2. Find the probability that the first day on which it rains during the event is Tuesday.
3. Find the probability that it rains on exactly one of the 4 days of the event.

6. One of the objectives of a computer game is to collect keys. There are three stages to the game. The probability of collecting a key at the first stage is \(\frac { 2 } { 3 }\), at the second stage is \(\frac { 1 } { 2 }\), and at the third stage is \(\frac { 1 } { 4 }\).

1. Draw a tree diagram to represent the 3 stages of the game.
2. Find the probability of collecting all 3 keys.
3. Find the probability of collecting exactly one key in a game.
4. Calculate the probability that keys are not collected on at least 2 successive stages in a game.

4. A bag contains 9 blue balls and 3 red balls. A ball is selected at random from the bag and its colour is recorded. The ball is not replaced. A second ball is selected at random and its colour is recorded.

1. Draw a tree diagram to represent the information. Find the probability that
   1. the second ball selected is red,
   2. both balls selected are red, given that the second ball selected is red.