Lesson Element
Using frequencies in tree diagrams to calculate
probabilities
Instructions and answers for teachers
These instructions should accompany the OCR resource ‘'Using frequencies in tree diagrams to calculate
probabilities’ activity which supports OCR Level 3 Certificate in Quantitative Problem Solving (MEI) and Level 3
Certificate in Quantitative Reasoning (MEI).
This activity offers an
opportunity for maths
skills development.
Associated materials:
'Using frequencies in tree diagrams to calculate probabilities’ Lesson Element learner activity sheet and
'Using frequencies in tree diagrams to calculate probabilities’ supporting PowerPoint.
June 2015
This resource was inspired by a page from David Speigelhalter’s website at
http://understandinguncertainty.org/using-expected-frequencies-when-teaching-probability which is a
great open resource with links and discussions. An accompanying PowerPoint file is provided for your
use and these instructions contain references to the relevant slides.
Introduction
In the OCR/MEI Quantitative Reasoning specification, it states that candidates should ‘be able to work
with a tree diagram when calculating or estimating a probability, including conditional probability.
Learners can choose whether to work with either frequencies or probabilities in tree diagrams.’. This
lesson element also considers using frequencies in tables and Venn diagrams as these can be a good
starting point for understanding their use in tree diagrams. (Slides 3, 4 and 5)
The idea of using frequencies is very simple: instead of saying ‘the probability of X is 0.20 (or 20%)’, we
would say ‘out of 100 situations like this, we would expect X to occur 20 times’. This simple reexpression can prevent confusion and make probability calculations easier and more intuitive for
learners by clarifying what a probability means. When we hear the phrase ‘the probability it will rain
tomorrow is 30%’, what do we mean? That it will rain 30% of the time? Over 30% of the area? In fact it
means that out of 100 such computer forecasts, we can expect it to rain after 30 of them. By clearly
stating what the ‘denominator’ is ambiguity is avoided. It has been shown that, by using frequencies,
people find it easier to carry out non-intuitive conditional probability calculations. This is the standard
approach taught to medical students for risk communication, and is used extensively in public
information, for example, in breast screening leaflets. (Slide 6)
These guidance notes should be used with the PowerPoint resource (Using frequencies in tree diagrams
to calculate probabilities) and the student worksheet which provides additional questions for learners to
try individually or in pairs/groups. The intention is that students will discuss the questions in pairs or
groups and present their methods to the rest of the group. They could also be asked to create similar but
different problems to set the group.
The slides can be used as back up to demonstrate approaches that the students do not come up with
themselves – for example, students may initially not be aware that using frequencies is a good method
or that they could use a Venn diagram or table.
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Task 1 Basic probability
This is essentially a one level tree. Questions can involve going either from probabilities (expressed as
decimals, fractions and %’s) to expected frequencies, or vice versa. The problems can be drawn as
either expected frequency or probability trees. Do not provide any guidance at first – use the questions
to assess prior learning and preferred methods and explanations.
•
Some balanced dice have probability
1
6
of coming up ‘4’. Out of 60 throws, how many ‘4’s would
we expect to come up? (Slides 7 and 8 show two approaches)
•
80% of the school students can roll their tongues. If I pick 1000 students at random, how many do
you expect will NOT be able to roll their tongues? (Slides 9 and 10 show two approaches)
•
In a typical school with 80 Year 10 students, 64 of them will have a profile on the social media site
‘Face-ache’. What is the probability that if we pick a Year 10 student at random, they will not have
a profile? (Slide 11 and 12 show a possible approach)
Task 2 Comparison of probabilities
This involves comparison of two different situations, and can be represented using a pair of trees. It is
ideal for dealing with challenging and realistic questions concerning relative and absolute risks.
Discussion questions – students work in pairs or groups and present alternative solutions:
•
A newspaper headline says that eating radishes doubles your chance of getting Smith’s Disease.
1% of people who don’t eat radishes get Smith’s Disease anyway.
o
Out of 200 people not eating radishes, how many would I expect to get Smith’s disease?
o
Out of 200 people eating radishes, how many would I expect to get Smith’s disease?
o
How many people have to eat radishes, in order to get one extra case of Smith’s disease?
(Slides 13 and 14 show two possible approaches)
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Task 3 Conditional probabilities
This requires two-level trees, and can also bring in two-way tables and Venn diagrams. First, given the
conditional probabilities, set up the frequency tree. Then calculate the expected frequencies and convert
back to probabilities if wanted.
Discussion questions – students work in pairs or groups and present alternative solutions:
•
Ask students to write a question based on the probability tree diagram below:
Possible question: A weather forecast is generally right. When it forecasts ‘rain’, 90% of the time it rains.
When it forecasts ‘no rain’, 70% of the time it does not rain. In a typical September they forecast rain on
two-thirds of days and no rain on one-third of days.
•
Draw a frequency tree diagram based on the information given.
•
How many days would you expect it to rain each September?
•
What is the probability that a random day in September is not rainy?
From the expected frequency tree, we expect it to rain on a total of 18 + 3 = 21 days in September, and
not rain in 9. So the probability that a random day in September is not rainy is
June 2015
9
30
= 0.3
Task 4 Reverse conditional probabilities
This is where things can get a bit tricky, but using expected frequency representations allows students to
tackle some of the classic non-intuitive probability problems – essentially Bayes theorem. If they can do
these, they have learnt a subtle and valuable skill.
Weather forecasting: of the times it rains, what proportion did the forecast get it right?
•
It rains 21 times, and in 18 the rain was forecast, so the proportion is
there is
6
7
18
21
=
6
7
: ie when it rains,
chance that the rain was forecast.
Using frequencies is essentially the same as, but easier to understand than, finding the same chance
using probabilities, and much easier than interpreting the formula P(A|B) =
P(A∩B)
P(B)
!
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© OCR 2015 - This resource may be freely copied and distributed, as long as the OCR logo and this message remain intact and OCR is acknowledged as the
originator of this work.
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June 2015