Risk, Expected Frequency and Probability Diagrams

A number you are meant to feel

A probability and a risk are the same kind of number used for two different purposes. When somebody tells you the probability of rain is 0.3, they are describing something. When somebody tells you a habit doubles your risk, they are trying to make you act. You have already met the chart version of this problem, where the same honest data can be drawn to alarm you or to reassure you. This is the same critical habit applied to numbers instead of pictures, and it comes down to two questions you should ask every time. First, how many people does this actually mean? A probability of 0.25 is abstract, but fifty people out of two hundred is something you can picture, and your specification asks you to express risk in exactly that way for exactly that reason. Second, doubled from what? A risk that doubles from very small to slightly less small has doubled, and is still very small. Those two questions are most of this topic, and they are also most of what makes somebody hard to mislead.

Which tier needs which part

This topic is split across the tiers, and knowing where the line falls saves you revising something you will not be asked. Check your own tier before you go on.

How many of them

The probability that a randomly chosen member of a club plays chess is 0.2. There are 300 members. How many would you expect to play chess?

Turn the probability into people

A leaflet says the probability of a certain outcome is 0.25. Which of these expresses the same thing as an expected frequency, and why is that worth doing?

• About 50 people in a group of 200, because a number of people is easier to picture than a decimal
• About 25 people in a group of 200, because 0.25 is 25
• A quarter, because that is what 0.25 means
• It cannot be done without knowing how many people there are

Doubled from what

Higher tier. Here is a hypothetical example with invented round numbers, and it is the whole reason your specification distinguishes the two kinds of risk. In one group, 2 people in 1000 have a certain outcome. In another, 4 people in 1000 do.

Which two make the risk clear

A report describes the same hypothetical finding as the card above. Select the TWO statements that give a reader enough to judge it.

• The risk rose from 2 in 1000 to 4 in 1000
• The risk doubled, from 0.2% to 0.4%
• The risk doubled
• There were twice as many cases in the second group

The claim that leaves out the baseline

If you take one thing from this module into the rest of your life, take this. Whenever you meet a claim that something doubles, triples or halves a risk, the claim is incomplete until you know the two numbers it is comparing. Doubling is a relationship, not a quantity. A risk that goes from 2 in 1000 to 4 in 1000 has doubled. So has a risk that goes from 200 in 1000 to 400 in 1000, and those two situations are nothing like each other. The first is a change most people would ignore; the second would change how you live. The relative figure cannot tell them apart, which is why it is the figure most often quoted on its own. In an exam, the mark is usually for saying what is missing rather than for calculating anything: a good answer states that the absolute risks are not given, explains why they are needed, and, if the numbers are there, converts them into expected frequencies so the reader can picture them. Notice that none of this requires you to accuse anyone of lying. Every individual statement can be true and the overall impression still misleading, which is exactly the point you have already met with charts.

The line that needs a baseline

Four sentences from a report about a hypothetical study. Select the ONE that a reader cannot judge without more information.

• In the first group, 2 people in every 1000 had the outcome.
• In the second group, 4 people in every 1000 had the outcome.
• The second group faced double the risk of the first.
• The difference amounts to 2 extra people in every 1000.

Reading the two-way table

A survey of 200 students records how they travel and what they eat at lunchtime. Of the 80 who come by bus, 48 bring a packed lunch. Of the 120 who walk, 72 bring one. So the number bringing a packed lunch altogether is _____, and the number using the canteen is _____. The probability that a student chosen at random brings a packed lunch is _____. The probability that a student chosen at random comes by bus is _____. And the probability that a student both comes by bus and brings a packed lunch is _____.

Match the diagram to its job

What the branches must do

You draw a tree diagram. From the first point, one branch is labelled 3/5. What must the other branch be, and why?

• 2/5, because the branches from any one point must account for everything that can happen and so sum to 1
• 3/5, because both branches are equally likely
• It could be anything, since it depends on the second event
• 1/5, because the two should differ

More trials, closer to the truth

A coin is spun and heads recorded. Put these three results in order, from the one furthest from the theoretical probability of 0.5 to the one closest.

• 7 heads in 10 spins, which is 0.700
• 54 heads in 100 spins, which is 0.540
• 508 heads in 1000 spins, which is 0.508

Explaining a risk honestly

A question gives you a hypothetical finding: in one group 2 people in every 1000 had a certain outcome, and in another group 4 people in every 1000 did. A newspaper reports that the risk has doubled. You are asked to comment on the report. Here is an answer that would score well. The report is accurate but incomplete. The relative risk is indeed 2, since 4 in 1000 is twice 2 in 1000, so it is true that the risk doubled. However, the absolute risks are 0.2% and 0.4%, and the difference between them is 2 additional people in every 1000. Expressing it that way makes the size of the change clear in a way the word doubled does not, because a doubling of a very small risk is still a very small risk. A reader given only the relative figure cannot tell this situation apart from one in which the risk rose from 200 in 1000 to 400 in 1000, which would be far more serious. The report would be improved by giving both absolute figures alongside the comparison. Notice what that answer does. It concedes that the claim is true before criticising it. It gives the arithmetic. It converts to expected frequencies so the reader can picture the change. And it says what would fix the report, rather than stopping at the complaint.

Advising on a headline

A student magazine wants to report a hypothetical study in which an outcome occurred in 2 of every 1000 people in one group and 4 of every 1000 in another. Take the decisions in order.

• The draft headline says the risk doubles. Is that true?
• It is true. So what is wrong with printing only that?
• You want to add one sentence that fixes it. What should it contain?
• An editor asks whether the original headline was dishonest. What is the accurate answer?

Explain the risk properly

A friend reads that a habit doubles the risk of something and is worried. Explain what they need to know before deciding how worried to be, using the ideas from this module.

• Explain what an expected frequency is and how to work one out from a probability
• Explain why expressing a risk as a number of people in a group is clearer than a decimal
• Explain the difference between relative risk and absolute risk
• Explain why "the risk doubles" is incomplete on its own, using an example of your own
• Show that a doubling of a very small risk is still a very small risk
• Say what a report should include so that a reader can judge the claim fairly
• Explain how experimental results get closer to the theoretical probability as trials increase
• Finish by saying what you would tell your friend to look for in the article