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You can use this random tool to estimate the areas of shapes. How?

Many statistical issues are raised by this activity, such as:

- How would you make the decision to stop making random samples to be confident of correctly predicting an area to 1 decimal place?
- Could you design an algorithm which could be used to estimate areas using this tool?
- How few trials would you need to be confident about the predictions of an area? Is this dependent on the type of shape or just the area of the shape?
- Could you use the tool to create a conjecture for the exact formula for the area of an ellipse?
- What sort of shape would the tool work best for? Worst for?

NOTES AND BACKGROUND

This tool is the basic idea behind Monte Carlo methods of integration, of which finding areas of shapes is a special case. Many integrals used in practice are too difficult to calculate exactly and can be time consuming to integrate numerically when they become sufficiently complex. The Monte Carlo method of integration is a quick and effective way to handle really difficult integrations. It is of great use in financial modelling.