Teaching notes
Teaching notes
Unit 16
16 Probability
Objectives
★ Use the language of probability to describe and interpret results involving likelihood and chance
★ Understand and use the probability scale from 0 to 1
★ Find probabilities based on equally likely outcomes in simple contexts
★ Identify all the possible mutually exclusive outcomes of a single event
★ Use experimental data to estimate probabilities
★ Compare experimental and theoretical probabilities in simple contexts
★ Record and explain methods, results and conclusions
Possible lessons
Lesson
Topic
Number of
40-minute
periods
Resources in
Coursebook
Resources in
Practice Book
1
The probability scale
1
Page 153
Page 76
2
Equally likely
outcomes
2
Pages 154–155
Page 77
3
Mutually exclusive
outcomes
1 or 2
Pages 156–157
Page 78
4
Estimating
probabilities
2
Pages 158–159
Pages 79–80
Resources in
Teacher’s Resource
Assumed prior knowledge
• Students may have some knowledge of using the language of probability to discuss events (stages 5 and 6) but it
is not an essential requirement.
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Unit 16
Teaching notes
Lesson 1 16.1 The probability scale
Coursebook page 150
Key words
likely
unlikely
certain
chance
even chance
very unlikely
probability
probability scale
Main teaching points
• Students need to know and be able to use words that describe the chance of any particular outcome of an event
occurring. These will include, in order of increasing likelihood, ‘impossible’, ‘very unlikely’, ‘unlikely’, ‘even
chance’ or ‘50-50’, ‘likely’, ‘very likely’ and ‘certain’.
• Probability is what mathematicians use to quantify the chance that an outcome will occur. It is a number in
the range from 0 to 1 and can be written as a fraction, a percentage or a decimal. Impossible outcomes have
probability 0 and certain outcomes have probability 1. It is convenient to picture probabilities on a number line
stretching from 0 to 1.
Common misunderstandings and misconceptions
• Students may have incorrect ideas about chance that need to be challenged. For example, they often think that
some numbers come up more than others when a fair dice is thrown, or that if a number has not occurred in a
long sequence of throws then its chance of occurring has increased.
• Probabilities must not be written in the form a in b. For instance, a probability of 1 can be written as 25% or
4
0.25 but not as 1 in 4.
Problem solving
• Aspects of the problem solving objective ‘record and explain methods, results and conclusions’ are covered in
the activity below.
Activity
• Hang a washing line or length of string across the classroom. Have ready a series of cards with events and
outcomes written on them, with pegs to attach the cards to the line. Say that at one end of the line (the left-hand
end when looking at it) an outcome is impossible and at the other end it is certain. Attach cards saying this.
• Distribute the cards to selected students, one at a time, and ask them to attach them where they think they
should go.
• Ask these students to give reasons to justify their choices. Check if other students in the class agree. Any who
disagree should be asked to give reasons. Use words such as ‘likely’, ‘very likely’, ‘unlikely’, ‘even chance’ or ‘50-50’
and so on.
• Possible outcomes are: it will rain tomorrow; it will be hot tomorrow; someone in the class will be absent
tomorrow or next week; a dice will show a six when it is thrown; a coin will come up heads; a named team or
person will win a particular sporting event. There might be events in the news, such as unusual weather or
earthquakes, that can be used.
• When several cards have been attached to the line, explain that outcomes like these often have a number, called
the probability, associated with them. This can take a value from 0 to 1. An impossible outcome has probability
0 and a certain outcome has probability 1.
• Ask students to assign a probability to each outcome on the line and to justify their answers. Encourage the use
of fractions, decimals and percentages. Sometimes, such as when throwing a dice, fractions are easiest. For other
outcomes, decimals or percentages seem more natural.
Comments on questions
• Q1 You could extend this by asking for examples to match the words that have not been used.
• Q3 This question could be extended by asking students how they think those probabilities are calculated. In
some cases, such as a and f, historical data can be used. In events involving the weather, computer models are
often used; d might be the result of an experimental planting of lots of bulbs. These ideas will be discussed again
later in the unit.
Homework
Practice Book page 76
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Teaching notes
Lesson 2
Unit 16
16.2 Equally likely outcomes
Coursebook pages 154–155
Key words
event
outcome
equally likely
equally likely outcomes
random
Main teaching points
• When all outcomes of an event are equally likely, it is possible to assign a probability based on this fact.
Common examples of equally likely outcomes include throwing a dice or taking a ball out of a bag without
looking. If there are 10 balls in a bag and four are red, the probability of taking a red ball is 4 = 2 or 0.4
10 5
or 40%.
• Terms such as a dice being ‘fair’, or taking the ball ‘at random’ indicate that the outcomes are equally likely.
Obviously if you look in the bag as you take the ball, the probability will change.
Common misunderstandings and misconceptions
• Students do not always think about whether the different possible outcomes are equally likely before calculating
probabilities. Emphasise the need to do this and give examples in which the outcomes are not equally likely, to
emphasise this point.
Problem solving
• The activity below gives students an opportunity to experience the problem solving objective ‘explain methods,
results and conclusions’. Encourage the students to describe their strategies and give reasons in support of them.
They may need help in explaining their ideas clearly but encourage them to do it in their own words. Ask other
students to say whether they agree or disagree that any given method is a good one. More details are given in the
description of the activity.
• Question 3 part b in the exercise provides another chance for students to explain and justify. Use this as a basis
for class discussion, after students have attempted the question. Ask students to suggest and criticise different
methods of choosing a sample.
• Question 5 is based on a common misconception and is a useful question to discuss with the whole class.
Encourage students to explain clearly why it is incorrect.
Activity
• Students should work in pairs. You will need one dice. Ask each pair to draw a 3 by 3 grid. Say that you are going
to throw a dice repeatedly and they are going to try to guess the numbers that will come up. Before you start,
ask them to put one number, from 1 to 6, in each of the nine cells. They do not need to include all the numbers
if they do not want to and obviously there will be repeats. Then throw the dice repeatedly. Call out the number
and record it each time. Students cross off one number, if they have it, for each throw of the dice. The winners
are the first to cross off all nine numbers. Play this several times.
• Ask students to suggest a good strategy for this activity. Ask them to justify their suggestions. It seems sensible
to put each number once. Challenge any suggestion such as: ‘You always get 6 more often’ by looking at the
evidence of the throws that they have recorded.
• Students need to understand that, in the long run, each number will come up with roughly the same relative
frequency, although in the short term there can be big variations. Find the relative frequency of each number,
frequency for that number
, in all the throws of the dice that you made while playing the game. The frequency
total number of thrrows
should be about 17% 16 each time.
()
• This result justifies using the idea that all numbers are equally likely to give the probability of each number
as 1 . This can then be extended to give the probability of an even number as 3 = 1 and other similar results. In
6
6
2
this case it is convenient to use fractions rather than percentages or decimals.
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Unit 16
Teaching notes
Comments on questions
• Q3b Various methods are possible. Putting all the names on pieces of paper in a container and taking one out
without looking should be random. Some students might suggest using a register. In that case, how can they be
sure the choice is fair?
• Q5 This provides a chance to reinforce the idea that all the outcomes must be equally likely, for probabilities to
be assigned in this way. You could discuss how the probabilities could be assigned in this example. There will be
some element of guesswork involved, based on the recent performances of each team and other possible factors,
such as which players will be involved.
• Q8 Students need to realise that the probability of each outcome changes, each time the teacher takes out a pen.
Encourage them to work through the problem carefully, to answer it successfully. Make sure that they answer
the question fully and do not simply calculate the probability of Deshi getting a red pen.
Homework
• Practice Book page 77
Lesson 3 16.3 Mutually exclusive outcomes
Coursebook pages 156–157
Key word
mutually exclusive
Main teaching points
• Students need to be able to identify when outcomes of a single event are mutually exclusive.
• A single event includes such things as throwing a dice, taking a ball out of a bag or tomorrow’s weather. Each of
these events can have different outcomes, such as an even number, a ball that is not red, or sunshine.
• Two outcomes are mutually exclusive if they cannot both occur simultaneously. Take throwing a single dice as
an example and consider these three outcomes:
A − getting a 2
B − getting an odd number
C − getting a number more than 3.
• Outcomes A and B are mutually exclusive. Outcomes A and C are mutually exclusive. Outcomes B and C are not
(they both occur if you throw a 5).
Common misunderstandings and misconceptions
• Students can be confused about the difference between an event and an outcome. The words are not always used
consistently but in this section it is important to emphasise the correct use. As an example, an event is throwing
a dice; the possible outcomes are scoring 2, more than 4, or an odd number.
Problem solving
• The activity below gives students an opportunity to explain methods, results and conclusions. If they can
explain an efficient strategy clearly then you can ask then for the maximum number of questions needed in the
numerical case. Again, ask for a justification of the answer. More details are given below.
Activity
• Use this activity to clarify what ‘mutually exclusive’ means and, at the same time, to ask students to think of a
good problem solving strategy.
• Let students play this game in pairs. One person chooses a whole number from 1 to 20 inclusive, then writes it
down and hides it so that there is no cheating. The other person then tries to identify the number, asking as few
questions as possible. The questions can only be answered ‘yes’ or ‘no’. Record the number of questions asked;
see who can do it with the fewest. Some students may use a strategy of asking whether it is more or less than a
particular number each time. This can be very efficient. To add an extra challenge, you could ask those students
to do it without using the words ‘more than’ or ‘less than’.
• A variation is to ask each person to think of someone in the room. Again, the challenge for the questioner is to
identify the person chosen in as few questions as possible.
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Teaching notes
Unit 16
• Ask students to describe suitable strategies. They should realise that a good question will divide the numbers or
people into two mutually exclusive groups of roughly equal size. Introduce the term ‘mutually exclusive’, used in
this context, so that students will understand what it means.
• Challenge students to explain why they should never need to ask more than five questions in the numbers
version of the game. (It is because, in the worst possible case, one question will reduce the possible values to 10,
a second question to 5, a third to 3, a fourth to 2 and the fifth will decide it. Often it will be fewer.) Ask students
to decide on the maximum number of questions in the ‘person’ version.
• Discuss possible categories in the ‘person’ version. These could include gender, wears glasses, age and
number of brothers and sisters. In the latter two examples, there are more than two classes but they are all
mutually exclusive.
Comments on questions
• Q3b This introduces the idea of three events that are all mutually exclusive. This means that at most one of the
three can occur.
• Q3c If outcome D occurs then outcome B must occur. This means that the probability of B is greater than or
equal to the probability of D.
• Q4 One way to solve part a is to draw a time line and to mark each of the outcomes with a point or a line. It
would be helpful to use a different colour for each outcome, for identification. This shows that a good diagram
can make a question easier to understand and to answer. An easy answer to part b is that the aeroplane is early,
on time or late.
Homework
• Practice Book page 78
Lesson 4 16.4 Estimating probabilities
Coursebook pages 158–159
Key words
estimate
experimental probability
theoretical probability
Main teaching points
• Students need to know how to use the results of an experiment or collected data to estimate the probability of
different outcomes of a single event by finding relative frequencies. They need to realise that the estimate can
change if more data are collected. This doesn’t happen in the case of probabilities calculated by using the concept
of equally likely outcomes. To distinguish between these two methods, introduce the terms ‘experimental’ and
‘theoretical’ probabilities.
• Sometimes it is possible to calculate both experimental and theoretical probabilities. In that case, it is interesting
to compare the two. You would not expect them to be exactly the same but when the differences are large it calls
into question the validity of the theoretical method (are the outcomes really equally likely?) or the accuracy of
the experiment.
• The drawing-pin example in the Coursebook is an example in which you cannot use equal likelihood to find a
theoretical probability (make sure students understand this point) and so experimental probability is the only
method available.
Common misunderstandings and misconceptions
• Students often do not make a clear distinction between the two different types of probability calculation. In
particular, they should realise that a theoretical probability is fixed and does not change. An experimental
probability depends on experimental data and always has a degree of uncertainty attached to it.
Problem solving
• The activity below provides opportunities for students to ‘explain methods, results and conclusions’. They
need to be able to decide on a method for testing the fairness of their spinner. They may decide to modify
the experiment in the light of their experience, perhaps by increasing the number of spins. There is no one
right answer here, so allow students to make their own decisions and carry them through. Give students the
opportunity to say what they think about other students’ decisions.
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Unit 16
Teaching notes
• The questions in the exercise give students the chance to consider results and conclusions and make their own
judgements about them.
Activity
• Ask students to make a hexagonal spinner and try to decide whether it is fair or not. They could work in pairs,
with one person spinning and the other recording. Each pair will need: isometric paper, card, scissors and a
cocktail stick or matchstick for the spindle.
• The easiest way to make the hexagonal shape required is to draw a hexagon on isometric paper, cut it out and
then use it as a template for a drawing on card. Students should write the integers from 1 to 6 on the outer edges
and push a stick through the centre. Then they can spin the spinner and record the number on the edge that
lands on the table.
• Discuss with students how they can decide whether the spinner is fair or not. Try to bring out the
following points.
* Some sort of experiment is required, involving spinning the spinner.
* The more spins the better.
* In the long run, if the spinner is fair, all six numbers will occur with roughly the same relative frequency.
* It will be useful to find the experimental probability for each number by working out
number of times the number is scored
total nu
umber of spins
and writing this as a decimal or percentage. Compare this to the theoretical value for a fair spinner of 1 or
6
about 17%.
• Allow students to decide how best to record their results. A tally chart would be a sensible choice. Suggest they
spin the spinner as many times as possible in the time allowed. Then they need to work out the experimental
probability for each number.
• Discuss the results from a selection of students. If possible, find one pair whose figures are very different from
the theoretical values and another whose values are very close. Which spinners are fair? Students can only use
their judgement to decide whether the spinner is fair or not. Formal statistical tests are beyond the scope of
this course!
Comments on questions
• Q5 You could extend this question by asking what we could find from more extensive records. For example, the
chance of rain at the start of the month might be different from the chance of rain at the end of the month. We
do not have the information to know that.
• Q6 Clearly the opponents for a particular match will change the probability here. This is an example in which
neither equal likelihood nor experiment can give the information needed to find the probability. Knowledge of
other factors will be required to make an informed estimate.
• Q8c Any points raised in the discussion of the activity above will be relevant here. If Raj said he scored exactly
20 heads and 20 tails we might not believe him − even though these are the ‘expected’ results.
• Q9 Here students are using the result of an experiment to make a judgement about the most likely situation
that gave rise to it. This is a technique that is used in medical diagnosis, or in fault diagnosis for complicated
machines such as aeroplanes. Given a set of symptoms, what is the most likely cause? Computer programs are
used to help make those judgements.
Homework
• Practice Book pages 79–80
6
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