Coin lines
If a coin rolls and lands on a set of concentric circles what is
the chance that the coin touches a line ?
Age
14 to 16
Challenge level
Problem
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- A coin is randomly tossed onto a set of parallel lines. If the line separation matches the diameter of the coin, what is the chance that the coin touches a line?
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And what if the line separation is different ? Double ? Half ?
- If the lines become concentric circles and the gap between the lines doubles, what is the chance now that the coin touches a line?
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An answer might be guessed at but
the problem is really about justifying that answer.
Do you have a way of visualising it and could you convince a friend?
If you succeed with that it should be an easy next step to generalise for a coin of any size.
Getting Started
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Find a coin, rule some lines on paper and experiment.
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Similarly create concentric circles with the correct separation, but this time keep a record of results.
Student Solutions
The person who sent in this solution didn't add their name, but it's a nice visualisation.
To start off, consider the situation where the distance between the two lines is double the diameter of the coin. Call the diameter of the coin d - so we can say the centre of the coin is 0.5d from the edge of the coin. The distance between the lines is 2d. For the coin not to cross either line, the centre of the circle must be a perpendicular distance of between 0.5d and 1.5d (inclusive) away from a line. The area between the two lines then "allowed" for the centre of the coin to land in without touching a line is therefore half of the total area between the two lines - so assuming the centre of the coin is equally likely to land in all areas, the coin will touch a line half the time.
Now a good way to think about the concentric circles is to imagine the coin somewhere and focus on a line through the centre of the coin and the centre of the concentric circles.
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The situation is the same as the one with straight lines I have considered already. So the answer is 0.5 or d, if the gap is 1 and the coin diameter is d.
Teachers' Resources
Why do this problem:
Probability often forces us to be particularly careful with our justification of answers. This problem has a simple enough numerical answer but the visualisation to support it must be carefully considered.Possible approach :
This problem might make a good poster, displayed somewhere it will catch students' attention to promote discussion.
The extent to which students need some practical activity will depend on how accustomed they are with visualisation tasks.Key questions :
- What do you think the answer might be?
- Do you have a way of looking at this situation so that you are sure your answer is right?
Possible extension :
- Research the problem context called Buffon's Needle
Possible support :
For students who cannot access this problem directly or theoretically the following activities may be helpful :- Draw some parallel lines at equal intervals and vary the size of that interval. Include in particular double and treble the coin diameter. Keep a tally of results.
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Draw the concentric circles and collect experimental data.