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Runs, strings and alphabets

Published online by Cambridge University Press:  23 January 2015

Barry Lewis
Barry Lewis*
Affiliation:
110 Highgate Hill, London N6 5HE, e-mail: barry@mathscounts.org
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Extract

In [1] Deshpande and Shiwalkar explore the number of double heads that may appear in sequences of coin tosses. They sought to answer questions such as,

‘If a (fair) coin is tossed r times, how many possible outcomes result in 0 double heads, how many result in 1 double head, …, how many result in (r – 1) double heads?’

Their answer involves Fibonacci expressions. Hirschhorn [2] extended these results using generating functions which on expansion led to some closed form expressions and the statistics of their distribution. Griffiths [3] continues this theme. Another article that touches on the basic idea – a run of particular outcomes – is [4] which is also the source of other references; [5] is also concerned with the same idea.

Type
Articles
Information
The Mathematical Gazette , Volume 98 , Issue 542 , July 2014 , pp. 204 - 226
Copyright
Copyright © Mathematical Association 2014 

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References

1. Shiwalkar, J. P. and Deshpande, M. N., The number of HHs in a coin tossing experiment and the Fibonacci sequence, Math. Gaz, 92 (March 2008) pp. 147–150.Google Scholar
2. Hirschhorn, M., Feedback on 92.27, Math. Gaz. 93 (March 2009) pp. 151–154.Google Scholar
3. Griffiths, M., Fibonacci expressions arising from a coin-tossing scenario involving pairs of consecutive heads, Fibonacci Quarterly 49 (August 2011) pp. 249–254.Google Scholar
4. Villarino, M. B., The probability of a run, Math. Gaz. 91 (March 2007) pp. 134–137.Google Scholar
5. Weisstein, E. W., Run, WolframMathWorld, accessed March 2014 at http://mathworld.wolfram.comlRun.html Google Scholar
6. Birkoff, G. and MacLane, S., A survey of modem algebra, Macmillan, 1954.Google Scholar
7. Camina, A. and Lewis, B., Introduction to enumeration, Springer-Verlag, 2011.Google Scholar
8. Graham, R., Knuth, D., and Patashnik, O., Concrete Mathematics, Addison-Wesley, 1998.Google Scholar
9. Noe, Tony, Tito, Oiezas III and Weisstein, E., Tribonacci numbers, WolframMathWorld, accessed March 2014 at http://mathworld.wolfram.comlTribonacciNumber.html Google Scholar