A Variation on Coin Tossing
Experiments

P. J. BUTT

Toss a coin N times recording the sequence of Heads and Tails obtained, then draw a graph by Plotting a ‘Head’ as one unit up and a ‘Tail’ as one unit down, against the Throw No. So for N = 10 and the sequence THHHTTTHTT the graph is:


 

Some investigations that can be carried out depending on the level of the pupils:

1. Finishing Position. On the graph it is at -2. What are the modal and mean finishing positions, and what is the distribution?

2. The number of times the graph is above the Throw No. axis. On the graph it is three times. What are the modal and mean number of times, and what is the distribution?

3. The number of times that the Throw No. axis is crossed. On the graph this happens twice. What is the mode and mean number of times, and what is the distribution?
 
 

Comments: Ten tosses of a coin appears to be a reasonable number, for this allows pupils to obtain a few sequences, and be able to draw the graphs in a single lesson.

It is worth asking the pupils to predict what the answers will be before obtaining the class’s overall result. The predictions are usually correct for (1), which is reassuring because the results can be disturbing for (2) and (3). With able and senior pupils, interesting discussions on the so called ‘Law of Averages’ can result when comparing the mode and mean in (2) and considering the mode and its frequency (3).

The central limit theorem can be demonstrated impressively using (2), for the ‘U’ distribution is changed into a ‘bell’ distribution, a near reversal of shape.

Results: For N = 10 there are 210 possible sequences.
 
 
(1) A binomial distribution where n = 10 and p = 1/2, results only occurring at even numbered positions.

(2)
 
No. of times above axis
0
1
2
3
4
5
6
7
8
9
10
  f
252
126
70
70
60
60
60
60
70
70
126

(3)
 
No. of crossings
0
1
2
3
4
  f
504
336
144
36
4

City of London School

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