Probability Trees
In this unit we introduce the idea of probability trees. This device is then used to calculate the theoretical probability of a number of events in a variety of situations
find probabilities of events using probability trees invent their own situations and ask probability questions in these contexts
Probability plays an important role in both the private and public sectors of modern life. Most of us play games of chance in some way or other. This may vary from a simple board game to playing games in a casino for money. In corporate life, actuaries are employed to assess risk of almost every kind so that insurance companies make profits on the premiums they set. There are a number of ways to try to determine the probability of an event. Here we use probability trees. This method is useful when we have basic activities with known theoretical probabilities, such as rolling a dice, tossing a coin, drawing a card from a deck or taking a ball from a barrel. When these activities are repeated we can find the probability of an event such as getting three sixes in three rolls of a dice, by constructing a probability tree. In a later unit for Level 6, we will go on to consider the use of probability trees in situations where two different basic events are combined and to compute conditional probability. But here we will just be concerned with learning how to construct and use probability trees when one activity is repeated. Along the way we will also provide students with the opportunity to look for patterns in their work. Here we will look for generalisations. What will happen when a coin is tossed n times? What is the likelihood of getting precisely one Tail? Two?These generalisations are basically like finding algebraic patterns but involve slightly more sophistication. Part of the reason for looking for these generalisations is that this is an important part of problem solving and one of the ways that mathematics develops. The more ways that a student can be induced to think about a problem the better will their understanding be of the mathematics behind the problem. So we not only give problems that require numeric answers but we give numeric answers that require problems. We also provide the opportunity for students to produce their own problems. By doing this they will see better what is going on in a given situation. They will also see how to specify more precisely what they are asking and so become aware of such subtleties as the difference between at least two Heads and precisely two Heads occurring. It might be worth pointing out that students should be familiar with the addition and multiplication of fractions before they start this unit. They will need to be able to simplify fractions such as 3/27 and add and multiply fractions like 1/8, 1/16, and 4/13
Session 1
In this session the main aim is to revise the students’ knowledge of probability and start to think about ways to solve particular probability problems. This will lead to a tree approach to counting that will develop into probability trees in the next session.
Teachers’ Notes
There are three elements to what is done here: solving a problem; looking for a general method that might work with other problems; and extending the method to work with other problems. Along the way, develop any generalisations that you feel able to.
Teaching Sequence
There are four sequences of questions in this session. If there is not enough material, then get the students to try variations. We have suggested some below. If there is too much material, then cut back on the generalisation side or delay some questions until the next session.
In all the questions of this session concentrate on first finding the number of occurrences of the required events and the number of total occurrences. Then use these to find the probability of the required event.
Begin the session with the whole class working together. Ask them some questions to get them started. This set of questions is aimed to get the students to recall what they know about probability. We give some possible answers in brackets. There are clearly many more answers. You might even ask them to say how they would make spinners to produce these probabilities.
Tell me something with a probability of 1/2. (Getting a head when a coin is tossed.)
Good, what else has a probability of 1/2? (That it’s night or day, approximately. Rolling an even/odd number with a dice.)
Can you think of something that happens with a probability of 1/6? (Rolling a 6 with a dice.)
What about something that happens with probability 2/6? (Rolling a 1 or 2 with a dice.)
Can you invent something that has a probability of 14/27? (Perhaps this is the probability of choosing someone in the room who has black hair.)
2. When everyone seems to have remembered what probability is all about, then move on to a more complicated situations. You may need to draw some pictures to check out what is happening with three or more coins.
What is the probability of getting a Tail if I toss a fair coin? (1/2.)
What is the probability of getting two Tails if I toss two fair coins? (1/4 – can you explain why?)
What is the probability of getting three Tails if I toss three fair coins? (1/8 – can you explain why?)
What is the probability of getting four Tails if I toss four fair coins? (1/16 – can you explain why?)
What is the probability of getting ten Tails if I toss ten fair coins? (2-10 – can you explain why?)
Can you generalise that? What is the probability of getting n Tails if I toss n fair coins? (2–n – can you explain why?)
3. In the next lot of examples, let the students guess if they like. Even record their guesses. But try to encourage them to provide reasons for their answers. Accept the answers whether they are right or wrong. Don’t give them the correct answers at this point.
What is the probability of getting two Tails if I toss two fair coins? (1/4)
What is the probability of getting exactly two Tails if I toss three fair coins? (3/8)
What is the probability of getting exactly two Tails if I toss four fair coins? (6/16)
What is the probability of getting exactly two Tails if I toss ten fair coins? (45/210)
What is the generalisation for this?
What is the probability of getting exactly two Tails if I toss n fair coins?(n(n-1)/2 x 2n)
4. Let the class go to work in their groups (of 2 to 4) to try to find the correct answers to these questions. If they know a way to get the answer, then ask them to find another way. The two ways that we have in mind are to make a systematic list and to use a probability tree.
The ways to get exactly two Tails with three tosses of a coin are
HTT; THT; TTH.
The ways to get exactly two Tails with four tosses of a coin are
HHTT; HTHT; HTTH; THHT; THTH; TTHH.
As we go through five tosses, six tosses etc., you can see that the numbers of ways of getting exactly two heads with n tosses of the coin are is n(n – 1)/2. This is the triangular numbers. The students should be able to guess that pattern. (You may know it as nC2 – the number of ways of choosing 2 things from n.)
Working from a tree diagram we get the same results.
So the students should be able to work out the answers to the first three questions using either of these methods. The 4th and 5th questions are intended to extend more able students.
5. After the groups have worked at the problems for a while, then bring the class back together and share their results.
6. Send the students back into their groups to try the following questions that are on Copymaster One.
What is the probability of getting three Tails if I toss three fair coins?
What is the probability of getting exactly three Tails if I toss four fair coins?
What is the probability of getting exactly three Tails if I toss five fair coins?
What is the probability of getting exactly three Tails if I toss ten fair coins?
What is the generalisation for this?
Make sure that the students understand how to do these questions using a tree diagram (or a systematic list).
It would be good if the students could get used to understanding what a generalisation is. It would be even better if they could make a guess as to the right answer. Encourage the better students to give some reason for their guess.
7. Bring the groups back together and let them discuss their answers. Give them time to write down what they have done and why. Give them notes if you think that they are unable to produce them for themselves.
Session 2
Introduce the idea of a probability tree and use it to find simple probabilities.
Teacher’s Notes
In this session we are going to look at questions similar to the ones in the last session but we will go straight to probabilities.
In working through the probabilities here we have looked at both the direct way and the indirect way of finding the probability. You may feel that this is taking too much on at one time. In that case avoid the indirect way at first. Then come back and use that approach later. The point of doing both of these methods is that the indirect way is sometimes quicker and it is a useful way to remind students that the sum of all probabilities is one.
Teaching Sequence
1. Recall the work that was done in the last session.
How can I find the probability of getting two Heads with two throws of a coin?
Discuss the methods used in the last session. Concentrate on the tree diagram. Talk about how the probabilities can be added to this diagram to save time calculating. (See the diagram below.)
The answer is just 1/4.
Start to use the name probability tree for this diagram. It clearly is about probabilities and it looks a bit like a tree.
What is the probability of getting a Head and a Tail when I toss two coins?
Here they will need to see that the event occurs twice and that each occurrence has a probability of 1/4. So that the total probability is 1/4 + 1/4 = 1/2.
How can I find the probability of getting at least one Head when I toss two coins?
The probability of this can be found directly by adding the probabilities of getting at least one head. It can also be found by finding the probability of not getting a head and subtracting this from 1. Both answers are the same: 3/4.
What is the probability of getting exactly two Heads when I toss three coins?
This can be determined directly from the probability tree below.
What is the indirect way of getting this probability? How can we get it using (1 – some other probability)?
Discuss this with them.
2. Let the students work in their groups on Copymaster 2.1. As they are working move around the class and give help where it is needed. If there is any particular point at which more than one group is getting stuck, then call the class together and discuss the difficulty with them.
Copymaster 2.1
Suppose three coins are tossed. Use a probability tree to find the following probabilities.
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The probability of getting three Heads.
- The probability of getting exactly two Heads.
- The probability of getting at least two Tails.
- The probability of getting more Tails than Heads.
- The probability of getting fewer Tails than Heads.
- The probability of getting as many Heads as Tails.
The purpose behind this piece of work is just to allow them to practice using the probability trees.
The answers here are: 1. 1/8 2. 3/8 3. 4/8 = 1/2 4. 4/8 = 1/2 5. 4/8 = 1/2 6. 0.
It’s worth noting that, because of the symmetry of Heads and Tails in this situation, the probability of getting more Tails than Heads equals that of getting more Heads than Tails.So the answer for 5. can be obtained directly from the answer for 4.
3. If you think that some of Copymaster 2.1 needs to be discussed as a whole class bring the groups together and go through the work with them. It may be worthwhile getting them to write a summary of this work in their books.
4. In their groups, let them go on to Copymaster 2.2.
Copymaster 2.2
A. Consider rolling two coins.
What events have probabilities 1/4; 2/4; and 3/4?
B. Consider rolling three coins.
What events have probabilities 1/8; 2/8; and 4/8?
C. Consider rolling four coins.
What events have probabilities 1/16; 2/16; and 5/16?
D. Consider rolling n coins.
Can you generalise any of the probabilities above?
The purpose of this Copymaster is for the students to use what they know about probability trees to explore a new situation.
The kinds of answers we expect are:
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The events that get a probability of 1/2n are all Heads or all Tails. Every other event occurs more than once.
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The only events that occur twice are all Heads or all Tails.
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There are two events that give a probability of (n+1)/ 2n. These are {at most one Head}; {at most one Tail}.
5. Bring the class together and discuss the results of the work on Copymaster 2.2. Let them write up appropriate notes in their books.
Session 3
This session will go over more than a lesson. Here the students work in their groups on different sets of problems. The aim of these stations is to practice their use of probability trees in various situations.
Teacher’s Notes
This session we have provided a considerable amount of material. It is possible that the class will not be able to cover it all in two lessons. We suggest that you give them the problems a set at a time and discuss the answers in between each set.
You could also get them to note the way that the problem sets change. In fact you could ask them how the sets might change. In fact we started with a simple probability tree that had two possibilities at each stage. We either had Heads or Tails. So one way to change things is to have more than one possibility. This we do in Spinners 1 where we have three alternatives.
In the Heads and Tails scenario, each event (Head or Tail) was equally likely. The same thing occurs in Spinners 1. Another variation is to allow basic events to have different probabilities. In Spinners 2, getting the blue region is twice as likely as getting the red or white regions.
The two balls in barrels scenarios first allow just two basic events but different probabilities and then three basic events with different probabilities.
The Two Suits scenario is complicated by the fact that thee are 13 basic events. This is actually not a very good example to use probability tress on. This is because it’s very hard to draw a tree with 13 branching points.
The other variation that occurs right from the start is in the number of ‘levels’ of the tree – how many times the basic events are used. Do you draw two cards or three or more? We have limited this drawing in session 3 largely because it gets very complicated to draw a tree that has more than three levels.
Teaching Sequence
1. Recall the basic ideas involved with probability trees.
What aspects of probability trees can you change?
2. Give everyone a copy of Copymaster 3.1 and let them work on it together in their groups. Go around and give assistance where it is necessary. You might even run a small tutorial with two or three groups who are stuck on the same point.
3. Early finishers can move on to the next Copymaster but when most people have completed Copymaster 3.1, get the whole class to think about the answers together. Allow different students to present their results. Discuss and resolve any differences.
4. Continue in the above way until you have done enough of the Copymasters as you think the class needs in order to understand the idea behind probability trees.
5. Determine what work needs to be recorded in their books and allow time for this.
Copymaster 3.1 Spinners 1
This activity involves probability with a spinner. The three parts of the spinner are equal in area but are coloured red, white and blue.
A. If the spinner is spun twice, draw up the probability tree for each outcome. Answer the following questions.
1. What is the probability of getting two reds?
2. What is the probability of getting a red and a white?
3. What is the probability of getting at least one blue?
4. What is the probability of getting two different colours?
5. What is the probability of getting white at least once?
6. What is the probability of not getting blue at all?
7. What events have probability 1/3, 7/9 and 8/9?
B. If the spinner is spun three times, draw up the probability tree for each outcome. Answer the following questions:
1. What is the probability of getting exactly two reds?
2. What is the probability of getting a red and two whites?
3. What is the probability of getting at least one blue?
4. What is the probability of getting two different colours?
5. What is the probability of getting white at least once?
6. What is the probability of not getting blue at all?
7. What events have probability 1/3, 4/9 and 2/9?
The answers here come from the probability trees. The answers can virtually be read off of the probability tree.
Part A. 1. 1/9 2. 2/9 3. 5/9 4. 6/9 = 2/3 5. 5/9
6. 4/9 (= 1 – probability of at least one blue = 1 – 4/9)
7. 1/3 = 3/9 – both colours the same; 7/9 – not a red and a white; 8/9 – not two reds.
Part B.1. 6/27 2. 3/27 = 1/9 3. 19/27 4. 19/27 5. 19/27 6. 8/27
7. 1/3 = 9/27 = 1 – 18/27 – 18/27 comes from any colour appearing just twice, so 9/27 from no colour appearing exactly twice; 4/9 – red occurs precisely once; 2/9 – all three colours occur.
Copymaster 3.2 Spinners 2
This activity involves probability with a spinner. The red and white parts of the spinner are equal in area. The blue part of the spinner is twice as big as the red and white parts.
A. If the spinner is spun twice, draw up the probability tree for each outcome. Answer the following questions:
1. What is the probability of getting two reds?
2. What is the probability of getting a red and a white?
3. What is the probability of getting at least one blue?
4. What is the probability of getting two different colours?
5. What is the probability of getting white at least once?
6. What is the probability of not getting blue at all?
B. If the spinner is spun three times, draw up the probability tree for each outcome. Answer the following questions.
1. What is the probability of getting three reds?
2. What is the probability of getting three blues?
3. What is the probability of getting three different colours?
The answers here come from the probability trees. The answers can virtually be read off of the probability tree.
Part A. 1. 1/4 x 1/4 = 1/16 2. 2(1/16) = 1/8 3. 1/2 + 1/8 + 1/8 = 3/4
4. 1/16 + 1/8 + 1/16 + 1/8 + 1/8 + 1/8 = 10/16 = 5/1;
5. 1/16 + 1/4 + 1/8 = 7/16 6. 1/16 + 1/16 + 1/16 + 1/16 = 1/4.
Part B. 1. 1/4 x 1/4 x 1/4 = 1/64 2. 1/2 x 1/2 x 1/2 = 1/8
3. 1/32 + 1/32 + 1/32 + 1/32 + 1/32 + 1/32 = 6/32 = 3/16.
Copymaster 3.3 Green and yellow balls
There are five balls in a barrel. Three of them are green and the others are yellow.
A. A ball is picked out of the barrel. It is then put back and another ball is chosen. Find the answers for the probabilities of the different situations.
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What is the probability of choosing two green balls?
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What is the probability of choosing two yellow balls?
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What is the probability of choosing a ball of each colour?
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What is the probability of choosing at least one yellow?
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What is the probability of choosing at least one green?
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What event has a probability of 13/25?
B. A ball is picked out of the barrel. It is then put back and another ball is chosen. This is put back and another ball chosen. Find the answers for the probabilities of the different situations.
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What is the probability of choosing three green balls?
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What is the probability of choosing three yellow balls?
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What is the probability of choosing precisely two green balls?
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What is the probability of choosing precisely two yellow balls?
The answers here come from the probability trees. The answers can virtually be read off of the probability tree.
Part A. 1. 3/5 x 3/5 = 9/25
2. 2/5 x 2/5 = 4/25
3. 3/5 x 2/5 + 2/5 x 3/5 = 12/25
4. 6/25 + 6/25 + 4/25 = 16/25 (or 1 – 9/25)
5. 9/25 + 6/25 + 6/25 = 21/25 (or 1 – 4/25).
6. 13/25 – choose two balls of the same colour.
Part B. 1. 3/5 x 3/5 x 3/5 = 27/125
2. 2/5 x 2/5 x 2/5 = 8/125
3. 3/5 x 3/5 x 2/5 + 3/5 x 2/5 x 3/5 + 2/5 x 3/5 x 3/5 = 54/125
4. 3/5 x 2/5 x 2/5 + 2/5 x 3/5 x 2/5 + 2/5 x 2/5 x3/5 = 36/125.
Copymaster 3.4 Green and yellow and brown balls
There are twelve balls in a barrel. Four of them are green; Six of them are yellow; and 2 of them are brown.
A. A ball is picked out of the barrel. It is then put back and another ball is chosen. Find the answers for the probabilities of the different situations.
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What is the probability of choosing two green balls?
- What is the probability of choosing a yellow and a brown?
- What is the probability of choosing two balls of different colours?
- What is the probability of choosing at least one yellow?
- What is the probability that both balls are the same colour?
B. A ball is picked out of the barrel. It is then put back and another ball is chosen. This is put back and another ball chosen. Find the answers for the probabilities of the different situations.
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What is the probability of choosing three green balls?
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What is the probability of choosing precisely two green balls?
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What is the probability of choosing precisely two yellow balls?
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What is the probability of choosing at least two green balls?
The answers here come from the probability trees. The answers can virtually be read off of the probability tree.
Part A. 1. 1/3 x 1/3 = 1/9
2. 2(1/2 x 1/6) = 1/6
3. 2(1/3 x 1/2 + 1/2 x 1/6 + 1/3 x 1/6)= 11/18
4. 1 – (1/2 x 1/2) = 1 – 1/4 = 3/4
5. 1/3 x 1/3 + 1/2 x 1/2 + 1/6 x 1/6 = 14/36 = 7/12.
Part B. 1. 1/3 x 1/3 x 1/3 = 1/27
2. 3(1/3 x 1/3 x 1/2 + 1/3 x 1/3 x 1/6) = 2/9
3. 3(1/2 x 1/2)(1/3 + 1/6) = 3/8
4. precisely two greens plus precisely three greens = 2/9 + 1/27 = 7/27
Copymaster 3.5 Two suits
Alice and Frank are playing with the cards from two suits. So they have the Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen and King of Hearts and the same cards in Spades. Ace counts as 1.
A. Alice draws a card and puts it back into the pack. Frank draws a card.
1. What is the probability that Alice and Frank both draw red cards?
2. What is the probability that Alice and Frank each draw an Ace?
3. What is the probability that Alice and Frank both draw even numbers?
4. What is the probability that Alice and Frank draw cards that sum to 10?
5. What is the probability that Alice and Frank both draw court cards?
6. What is the probability that Alice and Frank both draw numbers less than 5?
B.Alice draws a card and puts it back into the pack. Frank draws a card and puts it back into the pack. Alice draws another card.
1. What is the probability that Alice and Frank draw nothing but red cards?
2. What is the probability that Alice and Frank draw only Aces?
3. What is the probability that Alice and Frank just draw even numbers?
4. What is the probability that Alice and Frank only draw court cards?
5. What is the probability that Alice and Frank only draw numbers less than 5?
The answers here come from the probability trees. The answers can virtually be read off of the probability tree.
Part A. 1. 1/2 x 1/2 = 1/4
2. 1/13.1/13 = 1/169
3. 5/13 x 5/13 = 25/169
4. 9(1/13 x 1/13) = 9/169
5. 3/13 x 3/13 = 9/169
6. 4/13 x 4/13 = 16/169
Part B. 1. 1/2 x 1/2 x 1/2 = 1/8
2. 1/13 x 1/13 x 1/13 = 1/2197
3. 5/13 x 5/13 x 5/13 = 125/21997
4. 3/13 x 3/13 x 3/13 = 27/2197
5. 4/13 x 4/13 x 4/13 = 64/2197
Session 4
Give the students time to invent their own problems. Let them solve each other’s problems.
Teacher’s Notes
Students will understand sections of mathematics better if they look at the problems from various aspects and if they are involved in making up their own problems. This session allows them time to invent some problems that require probability trees.
It is hard to make up questions from scratch so you will need to remind them of the work that they have done and the scenarios they have looked at. Then together you might think about other scenarios and what questions you might ask in those settings. Give them the chance to work on one of these scenarios. However, if there is time, give them the opportunity to make up scenarios of their own.
It will probably be a good idea to limit the probability trees to two levels.
Teaching Sequence
1. Recall the problems that the class has been working on over the last couple of lessons. Let them tell you about the scenarios used, the questions solved, and the idea of the probability tree.
2. Discuss other possible scenarios.What probability questions might you be able to ask in these situations?
3. Let them work in their groups to write a scenario and some questions in that scenario. When this has been done, let the groups swap problem sets and solve each others problems.
4. This may require a certain amount of discussion between groups that have swapped problem sets as questions and scenarios may not be clear or well-defined. Allow the groups to resolve issues where possible.
5. Discuss answers.
6. Repeat the process but encourage each group to invent a totally new scenario this time. Also encourage them to spend a little longer making sure that their questions are precise.(You might like to award a chocolate frog for the best problem set and/or the best answers.)
7. Discuss the answers and give the class time to write up what you think is important from the session.
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