How Confident Are You?

Why do this problem?

This problem provides an experimental context in which to introduce the language of probability and to estimate probabilities, while investigating an interesting phenomenon - that in some cases, a crowd acting as individuals often make better decisions than the individuals of which it is made.

This also provides a context to discuss what is meant by a weather forecast that the chance of rain the next day is 70%.

Possible approach

Provide a transparent container which is full of small sweets or other small items - there should be too many for anyone to be able to estimate how many there are at all easily.
Tell the students to survey as many people as possible, asking them to make an estimate of the number of sweets in the container in the form:

 

Key questions

How many people gave a range which contained the correct answer?
What proportion do you think this should be?
How do you explain any discrepancies?
What does 9 chances out of 10 mean?  This is an opportunity to discuss the numerical probability scale.  

Possible extension

This technique can be applied to weather forecasts, climate change models, and much else.  The language of probability provides a way to indicate the degree of confidence which a prediction provides.

The Intergovernmental Panel on Climate Change has ranking scales:

Confidence Terminology	Degree of confidence of being right
Very high confidence	At least 9 out of 10 chance
High confidence	About 8 out of 10 chance
Medium confidence	About 5 out of 10 chance
Low confidence	About 2 out of 10 chance
Very low confidence	Less than 1 out of 10 chance

Likelihood Terminology	Likelihood of the occurrence/outcome
Virtually certain	> 99% probability
Extremely likely	> 95% probability
Very likely	> 90% probability
Likely	> 66% probability
More likely than not	> 50% probability
About as likely as not	33 to 66% probability
Unlikely	< 33% probability
Very unlikely	< 10% probability
Extremely unlikely	< 5% probability
Exceptionally unlikely	<1 % probability

Possible support

There are two aspects of this problem which may cause some difficult.  

One is the amount of data collected.  It may help students if they do any work by hand on a small sample of the data, and use a spreadsheet to deal with the complete data set.

The language of probability can be challenging conceptually - we often use language in an imprecise way, and don't think carefully enough about how we express ourselves.  Students should be encouraged to be precise in words when they talk about probabilities before moving into any calculation.