Activity: Dropping a Coin onto a Grid
A few hundred years ago people enjoyed betting on coins tossed on to the shock … would they cross a wrinkle or not ?
A man ( Georges-Louis Leclerc, the Count of Buffon, see “ Buffon ‘s Needle “ ) started thinking about this and worked out how to calculate the probability .
nowadays it is your twist to have a go !
You will need :
A small round coin ,
such as a US penny, a 1c Euro or 5 Rupee .
|A sheet of paper with a grid of 30 millimeter squares .|
- Measure the diameter of your coin: ____ mm
- a US Penny is 19mm, a 1c Euro is 16.25mm, a Rs 5 is 23mm
- Also measure the spacing of your grid (it may not print at exactly 30mm): ____ mm
- Put your sheet of paper on a flat surface such as a table top or the floor.
- From a height of about 5cm, drop the coin onto the paper and record whether it lands:
A : wholly inside a square ( not touching any grid lines )
bacillus : Crosses one or more lines
The accurate height from which you drop the coin is not important, but do n’t drop it so close to the newspaper that you are cheating !
If the mint rolls completely off the composition, then do not count that go .
now we will drop the mint 100 times, but first …
… what percentage do you think will land A, or B ?
Make a estimate ( estimate ) before you begin the experiment :
|Your Guess for “A” (%):|
|Your Guess for “B” (%):|
OK let’s begin .
Drop the coin 100 times and record A ( does not touch a line ) or B ( touches a line ) using Tally Marks :
now draw a Bar Graph to illustrate your results. You can create one at Data Graphs ( Bar, Line and Pie ) .
- Are the bars the same height?
- Did you expect them to be?
- How does the result compare with your guess?
We Can Calculate What It Should Be …
hera are some positions for the coin to land so it does not quite touch one of the lines :
Place your coin on your grid ( like above ), and then put a bell ringer on the wallpaper where the plaza of the coin is ( fair a harsh estimate will do ) .
See how the mint ‘s center is one radius r away from a line.
( Read about a Circle ‘s Radius and Diameter. )
Make lots of “ center marks ” then draw a box connecting them all like below :
d = Coin ‘s diameter ( 2 × roentgen )
When a coin ‘s center is within the yellow box it wo n’t touch any line .
The chicken box is smaller than the grid by two radiuses ( = one diameter ) of the mint .
So what are the areas ?
- The area of the grid square is 30 × 30 = 900 mm2
- The area of the yellow box is (30-d) × (30-d) = (30-d)2 mm2
The above calculation was for a 30 millimeter grid, but we can use S for power system size :
- The area of the grid square is S × S = S2 mm2
- The area of the yellow box is (S-d)2 mm2
Example: A 1c Euro (d=16.25 mm) on a 29mm grid (S=29 mm):
Grid Square = 292 = 841 mm2
yellow Box = ( 29-16.25 ) 2 = 12.752 = 162 mm2 ( to the nearest mm2 )
So you should expect the coin to land not crossing a line of the grid approximately :
“ A ” = 162 / 841 = 19.3 % of the time
And “ B ” = 100 % – 19.3 % = 80.7 %
now do the calculations for your own grid size and coin size .
|Grid Spacing second (mm):|
|Diameter of Coin five hundred (mm):|
|Area of Grid Square = S2 (mm2):|
|Area of Yellow Box = ( S-d ) 2 (mm2):|
How do these theoretical results compare with your experimental results ?
It wo n’t be claim ( because it is a random thing ) but it may be finale .
Different Sizes of Coin
Try repeating the experiment using a different sized mint .
- First calculate the theoretical value … how does this affect the values for A and B?
- Then do the experiment to see how close it gets.
What You Have Done
You have ( hopefully ) had fun running an experiment.
Read more: Mini Coin Purse – Free Crochet Pattern
You have done some geometry, and had some experience calculating areas and probabilities .
And you have seen the relationship between theory and world .