Unlucky lines
This is a level 4 statistics activity from the Figure It Out series. It is focused on describing a probability using fractions. A PDF of the student activity is included.
Figure It Out is a series of 80 books published between 1999 and 2009 to support teaching and learning in New Zealand classrooms.
This resource provides the teachers’ notes and answers for one activity from the Figure It Out series. A printable PDF of the student activity can be downloaded from the materials that come with this resource.
Specific learning outcomes:
- Describe a probability using fractions.
Unlucky lines
Achievement objectives
S4-4: Use simple fractions and percentages to describe probabilities.
Required materials
- Figure It Out, Level 4+, Statistics, Book Two, "Unlucky lines", page 20
- coins or counters
See Materials that come with this resource to download:
- Unlucky lines activity (.pdf)
Activity
In this activity, students find an experimental estimate of the probability of a coin landing between the lines on a grid and then try and discover the theoretical basis for their results. As always with practical experiments involving probability, they should realise the importance of obtaining a large number of results (the larger the better).
Collating results from a number of students is a good way of getting a large body of results in a short space of time.
Students’ explanations for the results of this experiment (question 1c) will vary according to their level of mathematical understanding. They will need good reasoning skills to justify their explanations mathematically. A full explanation is given in the Answers.
The students will probably want to use a trial-and-improvement approach to question 2. Starting in this way, they will soon realise that the mathematical solution is related to the area that can be occupied by a winning coin compared with the total area of each square in the grid.
Once they realise that the winning area is defined by the locus of the centre of the coin, the question becomes a measurement problem as much as a probability one. (See the diagram in the Answers.)
For your reference, this table shows how the probability changes as the size of the grid increases:
Side of square |
Win area |
Win area as a fraction |
Win area as a decimal |
|---|---|---|---|
40 mm |
289 mm2 |
289/1 600 |
0.18 |
50 mm |
729 mm2 |
729/2 500 |
0.29 |
60 mm |
1369 mm2 |
1369/3 600 |
0.38 |
70 mm |
2209 mm2 |
2209/4 900 |
0.45 |
80 mm |
3249 mm2 |
3249/6 400 |
0.51 |
Note that, although the concept of locus is not mentioned in the achievement objectives, it is included as a suggested learning experience for Level 4, Geometry.
1.
a. Practical activity.
b. Answers will vary, but it is likely that about 1/5 (0.2) of all attempts are successful.
c. The diameter of a $1 coin is 23 mm, so its centre is 11.5 mm (half of 23 mm) from its edge. This means that the centre of the coin can never be closer than 11.5 mm from the edge of the square, or it will touch. So no matter what size the squares of the grid are, there is always a “lose” area 11.5 mm wide along each side and a square “win” area is left in the middle. The diagram shows how this works for a grid with 4 cm squares:
The total area of a 4 cm by 4 cm square is 40 x 40 = 1 600 mm2. Of this, only the 17 mm by 17 mm grey square in the middle is the “win” area. 17 x 17 = 289 mm2.
This means that the chances of the centre of the coin landing on the winning area are:
- 289/1 600 = 0.18, or 18%.
- 2.80 mm x 80 mm, or very close to this. (78.5 mm, correct to the nearest 0.1 mm, will give a probability of 0.5.)
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