### To Insure or Not to Insure

Should you insure your mobile phone? It rather depends on whether you focus on the long-term pay-off or the effect of a single event.

Should Louis go for the safer options, hoping to limit his losses, or would he be better off with a riskier strategy, focusing on maximising his profit?

### Capture and Recapture

How do scientists or mathematicians estimate the size of a population of wild animals?

# How Confident Are You?

##### Stage: 3 and 4 Challenge Level:
How confident are you?

How confident are you in your guesses?

Fill a container with jelly beans (or similar small items) and ask as many people as possible to guess how many there are
- but don't just ask them for a simple guess!

Ask them to specify a range which they are 90% sure contains the right answer.

### 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%.

A 70% chance of rain for a particular location in a given time interval means that there are 7 chances in 10 that it will rain in that location in that time interval.

### 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:

I think there are 9 chances out of 10 that there are ... sweets in the jar.

Students should record the guesses, then display them as a range on a graph.  If desired, the mid-points of the ranges could also be displayed on a histogram.

### 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.