Monday, 11 July 2011

Galton's quincunx

Sir Francis Galton - born in Sparkbrook, Birmingham (UK) in 1822
was a cousin of Charles Darwin  

made contributions to many branches of Science

used a 'quincunx' (also called a Galton board or bean machine) to show how a distribution tends to a 'normal' curve

(original is in the Science museum, Boston (USA))

The purpose of this work is for students to compare (i) a short run experimental probability with (ii) a longer run and (iii) a theoretical value. Hopefully the longer run value will be more accurate.... The task also involves a systematic consideration of all options and hence the theoretical values (which can lead to Pascal's triangle).

Resources: students will need two copies (front and back) of the 8 quincunx sheet and a coin. It can be helpful for a pair to work together (one copy of the sheet between them). Later they may need the 16 quincunx sheet, when they sort out the theoretical outcomes.

Instructions: throw a coin (or roll a dice and look for odds and evens).
If it is a head: go right at a junction.
If it is a tail: go left.
This can be usefully demonstrated on a board.

Four flips of a coin should result in the ball ending up in one of the five bins. Students are asked to record 16 experiments on their own sheets (front and back).

“Why do you think it ends up in the middle bin more than the others?”

After this it might be helpful to consider the purpose of the lesson :
to compare:

  • A short run: your own experimental probabilities (their 16 trials)
  • A long run: collect and sum the frequencies for each box for each pair of students. Label the bins (1 to 5) and work out the overall experimental probabilities from the class results.
  • The theoretical probabilities: using another recording sheet ask students to find and/or draw all the different ways of getting from the top to the bottom: as a path or as combinations of 4 letters e.g. LLRR (following discussion). There are 16 ways altogether so they will need to be systematic. 

    “How can we use symmetry to simplify finding all the options?”
    The theoretical results are a row of Pascal’s triangle: 1, 4, 6, 4, 1. These frequencies will be out of 16 and can be compared (e.g. as a decimal) with the ‘long run’ results by changing the fractions into decimals.

    Utilising ICT: see (on a good day) how long-run experiments tend towards a theoretical value. 
    There are several good versions of Galton’s quincunx on the net:
    • (Duncan Keith) (is good visually when made full size) as you approach a good number of trials, slow down the drop rate with the speed slider - then click on auto-drop to stop the flow. The frequencies just seem to keep on accumulating if you leave it to run. It can be enlarged (bottom left) for a whiteboard.  
    • David Little (Penn State University) colourful (uses Java) can be made to fill a board and is downloadable - find the bin frequencies using the left and right arrow keys when the experiment has been halted, also displayed as percentages 
    • maths is fun has a bar chart helpfully displayed alongside a frequency count as well as the pinboard - but is a little small for classroom display for 4 layers. Gave the following frequencies for 1000 goes:   58 , 261 , 358 , 270 , 53.

      “Why do you think experimental results should be more accurate when more trials are involved (a longer run)?”

      “What is the difference between an experimental (short run and long run) probability and a theoretical value?”

      Extension: students could consider theoretical values for other layers (5 then 6 etc) leading to Pascal’s triangle.

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