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In the Bag

Stage: 3 Challenge Level: Challenge Level:1

A number of you commented on the fact that the distribution we obtain from taking random samples may not correspond to the actual distribution of the marbles. For example, if a sample of 10 marbles does not contain a blue ball this does not mean that there is no blue in the bag. Shanell from Garden International School adopted the following strategy:

My strategy is a simple one based on averages. Firstly take three viewings, each time recording the differences.

Add up each difference and then divide the total by the number of viewings (in this case three)

Example

Viewing 1: 3r + 3b + 3g + 1y
Viewing 2: 2r + 1b + 3g + 4y
Viewing 3: 5r + 1b + 1g + 3y
Average: (10r +5b + 7g + 8y)/3 = 3r + 2b + 2g + 3y [note that we round each term to the nearest integer, and make sure our number of balls adds up to 10]

The important point is that the more viewings we average over, the more likely we are to get the correct distribution. This is a very important concept in probability - it's worth giving it some thought if it doesn't make sense yet. A good resource for seeing this phenomenon in action is Which Spinners?

Bearing this in mind, can you suggest what might be thequickest strategy for getting to 1000 points? You might decide to take a fixed number of samples each round before guessing. You'll have to decide what this fixed number is going to be. Remember, the more samples you take each round the better your chances of guessing correctly, but the more points you expend. Maybe you could collect some data in a spreadsheet.