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Coin tossing (Posted on 2004-06-11) Difficulty: 3 of 5
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?

See The Solution Submitted by Federico Kereki    
Rating: 3.6667 (6 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Exhaustion--1st and 2nd questions | Comment 14 of 25 |
(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
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HHHHHTTHTH HHH
HHHHHTTHTT HHH
HHHHHTTTHH TTT HHH
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HHHHTHHTTT TTT HHH
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TTHHTHTTTH TTT
TTHHTHTTTT TTT
TTHHTTHHHH HHH
TTHHTTHHHT HHH
TTHHTTHHTH
TTHHTTHHTT
TTHHTTHTHH
TTHHTTHTHT
TTHHTTHTTH
TTHHTTHTTT TTT
TTHHTTTHHH TTT HHH
TTHHTTTHHT TTT
TTHHTTTHTH TTT
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
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