I threw a coin n times, and never got three tails in a row. I calculated the odds of this event, and found out they were just about even; 50%-50%. How many times did I throw the coin?

A second question: what were the chances of having not gotten three heads in a row either?

(In reply to re(3): Markov chain solution - 2nd part by Charlie)

Here are all the possibilities for 10 coin tosses, annotated with whether three heads and/or three tails appear in a row in the sequence:

HHHHHHHHHH HHH 
HHHHHHHHHT HHH 
HHHHHHHHTH HHH 
HHHHHHHHTT HHH 
HHHHHHHTHH HHH 
HHHHHHHTHT HHH 
HHHHHHHTTH HHH 
HHHHHHHTTT TTT HHH 
HHHHHHTHHH HHH 
HHHHHHTHHT HHH 
HHHHHHTHTH HHH 
HHHHHHTHTT HHH 
HHHHHHTTHH HHH 
HHHHHHTTHT HHH 
HHHHHHTTTH TTT HHH 
HHHHHHTTTT TTT HHH 
HHHHHTHHHH HHH 
HHHHHTHHHT HHH 
HHHHHTHHTH HHH 
HHHHHTHHTT HHH 
HHHHHTHTHH HHH 
HHHHHTHTHT HHH 
HHHHHTHTTH HHH 
HHHHHTHTTT TTT HHH 
HHHHHTTHHH HHH 
HHHHHTTHHT HHH 
HHHHHTTHTH HHH 
HHHHHTTHTT HHH 
HHHHHTTTHH TTT HHH 
HHHHHTTTHT TTT HHH 
HHHHHTTTTH TTT HHH 
HHHHHTTTTT TTT HHH 
HHHHTHHHHH HHH 
HHHHTHHHHT HHH 
HHHHTHHHTH HHH 
HHHHTHHHTT HHH 
HHHHTHHTHH HHH 
HHHHTHHTHT HHH 
HHHHTHHTTH HHH 
HHHHTHHTTT TTT HHH 
HHHHTHTHHH HHH 
HHHHTHTHHT HHH 
HHHHTHTHTH HHH 
HHHHTHTHTT HHH 
HHHHTHTTHH HHH 
HHHHTHTTHT HHH 
HHHHTHTTTH TTT HHH 
HHHHTHTTTT TTT HHH 
HHHHTTHHHH HHH 
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HHHHTTHTHT HHH 
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HHHTHHHHHH HHH 
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HHHTHHHHTH HHH 
HHHTHHHHTT HHH 
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HHHTHHHTHT HHH 
HHHTHHHTTH HHH 
HHHTHHHTTT TTT HHH 
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HHHTHHTHTH HHH 
HHHTHHTHTT HHH 
HHHTHHTTHH HHH 
HHHTHHTTHT HHH 
HHHTHHTTTH TTT HHH 
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HHHTTTTTHH TTT HHH 
HHHTTTTTHT TTT HHH 
HHHTTTTTTH TTT HHH 
HHHTTTTTTT TTT HHH 
HHTHHHHHHH HHH 
HHTHHHHHHT HHH 
HHTHHHHHTH HHH 
HHTHHHHHTT HHH 
HHTHHHHTHH HHH 
HHTHHHHTHT HHH 
HHTHHHHTTH HHH 
HHTHHHHTTT TTT HHH 
HHTHHHTHHH HHH 
HHTHHHTHHT HHH 
HHTHHHTHTH HHH 
HHTHHHTHTT HHH 
HHTHHHTTHH HHH 
HHTHHHTTHT HHH 
HHTHHHTTTH TTT HHH 
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HHTHHTHHHH HHH 
HHTHHTHHHT HHH 
HHTHHTHHTH 
HHTHHTHHTT 
HHTHHTHTHH 
HHTHHTHTHT 
HHTHHTHTTH 
HHTHHTHTTT TTT 
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HHTHHTTHTH 
HHTHHTTHTT 
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HHTHTHHHTT HHH 
HHTHTHHTHH 
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HHTHTHHTTH 
HHTHTHHTTT TTT 
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HTHHHHHHHH HHH 
HTHHHHHHHT HHH 
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HTHHHHHHTT HHH 
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HTHHHHHTHT HHH 
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HTHHTHHHTT HHH 
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HTHHTHHTHT 
HTHHTHHTTH 
HTHHTHHTTT TTT 
HTHHTHTHHH HHH 
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HTHHTHTHTT 
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HTHHTTHTHH 
HTHHTTHTHT 
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HTHTTTTHHH TTT HHH 
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HTTHHHHHHH HHH 
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HTTHHTHHTH 
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TTHHTTTHTT TTT 
TTHHTTTTHH TTT 
TTHHTTTTHT TTT 
TTHHTTTTTH TTT 
TTHHTTTTTT TTT 
TTHTHHHHHH HHH 
TTHTHHHHHT HHH 
TTHTHHHHTH HHH 
TTHTHHHHTT HHH 
TTHTHHHTHH HHH 
TTHTHHHTHT HHH 
TTHTHHHTTH HHH 
TTHTHHHTTT TTT HHH 
TTHTHHTHHH HHH 
TTHTHHTHHT 
TTHTHHTHTH 
TTHTHHTHTT 
TTHTHHTTHH 
TTHTHHTTHT 
TTHTHHTTTH TTT 
TTHTHHTTTT TTT 
TTHTHTHHHH HHH 
TTHTHTHHHT HHH 
TTHTHTHHTH 
TTHTHTHHTT 
TTHTHTHTHH 
TTHTHTHTHT 
TTHTHTHTTH 
TTHTHTHTTT TTT 
TTHTHTTHHH HHH 
TTHTHTTHHT 
TTHTHTTHTH 
TTHTHTTHTT 
TTHTHTTTHH TTT 
TTHTHTTTHT TTT 
TTHTHTTTTH TTT 
TTHTHTTTTT TTT 
TTHTTHHHHH HHH 
TTHTTHHHHT HHH 
TTHTTHHHTH HHH 
TTHTTHHHTT HHH 
TTHTTHHTHH 
TTHTTHHTHT 
TTHTTHHTTH 
TTHTTHHTTT TTT 
TTHTTHTHHH HHH 
TTHTTHTHHT 
TTHTTHTHTH 
TTHTTHTHTT 
TTHTTHTTHH 
TTHTTHTTHT 
TTHTTHTTTH TTT 
TTHTTHTTTT TTT 
TTHTTTHHHH TTT HHH 
TTHTTTHHHT TTT HHH 
TTHTTTHHTH TTT 
TTHTTTHHTT TTT 
TTHTTTHTHH TTT 
TTHTTTHTHT TTT 
TTHTTTHTTH TTT 
TTHTTTHTTT TTT 
TTHTTTTHHH TTT HHH 
TTHTTTTHHT TTT 
TTHTTTTHTH TTT 
TTHTTTTHTT TTT 
TTHTTTTTHH TTT 
TTHTTTTTHT TTT 
TTHTTTTTTH TTT 
TTHTTTTTTT TTT 
TTTHHHHHHH TTT HHH 
TTTHHHHHHT TTT HHH 
TTTHHHHHTH TTT HHH 
TTTHHHHHTT TTT HHH 
TTTHHHHTHH TTT HHH 
TTTHHHHTHT TTT HHH 
TTTHHHHTTH TTT HHH 
TTTHHHHTTT TTT HHH 
TTTHHHTHHH TTT HHH 
TTTHHHTHHT TTT HHH 
TTTHHHTHTH TTT HHH 
TTTHHHTHTT TTT HHH 
TTTHHHTTHH TTT HHH 
TTTHHHTTHT TTT HHH 
TTTHHHTTTH TTT HHH 
TTTHHHTTTT TTT HHH 
TTTHHTHHHH TTT HHH 
TTTHHTHHHT TTT HHH 
TTTHHTHHTH TTT 
TTTHHTHHTT TTT 
TTTHHTHTHH TTT 
TTTHHTHTHT TTT 
TTTHHTHTTH TTT 
TTTHHTHTTT TTT 
TTTHHTTHHH TTT HHH 
TTTHHTTHHT TTT 
TTTHHTTHTH TTT 
TTTHHTTHTT TTT 
TTTHHTTTHH TTT 
TTTHHTTTHT TTT 
TTTHHTTTTH TTT 
TTTHHTTTTT TTT 
TTTHTHHHHH TTT HHH 
TTTHTHHHHT TTT HHH 
TTTHTHHHTH TTT HHH 
TTTHTHHHTT TTT HHH 
TTTHTHHTHH TTT 
TTTHTHHTHT TTT 
TTTHTHHTTH TTT 
TTTHTHHTTT TTT 
TTTHTHTHHH TTT HHH 
TTTHTHTHHT TTT 
TTTHTHTHTH TTT 
TTTHTHTHTT TTT 
TTTHTHTTHH TTT 
TTTHTHTTHT TTT 
TTTHTHTTTH TTT 
TTTHTHTTTT TTT 
TTTHTTHHHH TTT HHH 
TTTHTTHHHT TTT HHH 
TTTHTTHHTH TTT 
TTTHTTHHTT TTT 
TTTHTTHTHH TTT 
TTTHTTHTHT TTT 
TTTHTTHTTH TTT 
TTTHTTHTTT TTT 
TTTHTTTHHH TTT HHH 
TTTHTTTHHT TTT 
TTTHTTTHTH TTT 
TTTHTTTHTT TTT 
TTTHTTTTHH TTT 
TTTHTTTTHT TTT 
TTTHTTTTTH TTT 
TTTHTTTTTT TTT 
TTTTHHHHHH TTT HHH 
TTTTHHHHHT TTT HHH 
TTTTHHHHTH TTT HHH 
TTTTHHHHTT TTT HHH 
TTTTHHHTHH TTT HHH 
TTTTHHHTHT TTT HHH 
TTTTHHHTTH TTT HHH 
TTTTHHHTTT TTT HHH 
TTTTHHTHHH TTT HHH 
TTTTHHTHHT TTT 
TTTTHHTHTH TTT 
TTTTHHTHTT TTT 
TTTTHHTTHH TTT 
TTTTHHTTHT TTT 
TTTTHHTTTH TTT 
TTTTHHTTTT TTT 
TTTTHTHHHH TTT HHH 
TTTTHTHHHT TTT HHH 
TTTTHTHHTH TTT 
TTTTHTHHTT TTT 
TTTTHTHTHH TTT 
TTTTHTHTHT TTT 
TTTTHTHTTH TTT 
TTTTHTHTTT TTT 
TTTTHTTHHH TTT HHH 
TTTTHTTHHT TTT 
TTTTHTTHTH TTT 
TTTTHTTHTT TTT 
TTTTHTTTHH TTT 
TTTTHTTTHT TTT 
TTTTHTTTTH TTT 
TTTTHTTTTT TTT 
TTTTTHHHHH TTT HHH 
TTTTTHHHHT TTT HHH 
TTTTTHHHTH TTT HHH 
TTTTTHHHTT TTT HHH 
TTTTTHHTHH TTT 
TTTTTHHTHT TTT 
TTTTTHHTTH TTT 
TTTTTHHTTT TTT 
TTTTTHTHHH TTT HHH 
TTTTTHTHHT TTT 
TTTTTHTHTH TTT 
TTTTTHTHTT TTT 
TTTTTHTTHH TTT 
TTTTTHTTHT TTT 
TTTTTHTTTH TTT 
TTTTTHTTTT TTT 
TTTTTTHHHH TTT HHH 
TTTTTTHHHT TTT HHH 
TTTTTTHHTH TTT 
TTTTTTHHTT TTT 
TTTTTTHTHH TTT 
TTTTTTHTHT TTT 
TTTTTTHTTH TTT 
TTTTTTHTTT TTT 
TTTTTTTHHH TTT HHH 
TTTTTTTHHT TTT 
TTTTTTTHTH TTT 
TTTTTTTHTT TTT 
TTTTTTTTHH TTT 
TTTTTTTTHT TTT 
TTTTTTTTTH TTT 
TTTTTTTTTT TTT 

Statistics are:

TTT 520 ; HHH 520 ; both 194 ; neither 178

So indeed, as we agree, (1024-520)/1024 = 49.2% do not have three tails in a row.

Of the 1024-520 = 504 that do not have three tails in a row, 178 also do not have three heads in a row (they have neither). Then 178/504 = 35.3% -- not 29.6%.

Posted by Charlie on 2004-06-13 11:25:55