Coin tossing sequences Martin

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Presentation transcript:

Coin tossing sequences Martin

Toss a coin repeatedly until we get a particular sequence. e.g. HTT TTTHHTHTT 9 tosses How many tosses on average? Is it the same for all sequences?

How many tosses on average to get HTT or HTH? Options: A.HTT takes longer B.HTH takes longer C.Both the same THT THH H T

HHTHTTTTTHHH HTTTHTHHTTTT HHTHTHHTTTHT HTTHHTHTTTHT Probability = 1/8 Average wait = 8 Target: HTT

HHTHTTTTTHHH HTTTHTHHTTTT HHTHTHHTTTHT HTTHHTHTTTHT Probability = 1/8 Average wait > 8 Target: HTH Overlaps don’t get counted

Calculating the average wait Bet £1 on each coin being start of chosen sequence THHTTHTH Target: HTH Stake Payout In a fair game, payout for matching all 3 is £8 Also payout £4 for matching 2, £2 for matching 1 For HTH, payout = £10 For fair game, average stake = payout  Average number of tosses = 10 for HTH

Calculating the average wait Bet £1 on each coin being start of chosen sequence THTTHTH Target: HTH Stake (£) Payout (£) In a fair game, payout for matching all 3 is £8 Also payout £4 for matching 2, £2 for matching 1 For HTH, payout = £10 For fair game, average stake = payout  Average number of tosses = 10 for HTH

Average wait Sequence length n Minimum wait = 2 n Maximum wait = 2 n+1 -2 Longer sequence always has longer wait SequenceAverage wait HHH14 HHT8 HTH10 HTT8 THH8 THT10 TTH8 TTT14 SequenceAverage wait HH6 HT4 TH4 TT6 SequenceAverage wait H2 T2

Two sequences Which is more likely to occur first? e.g. HTT, HTH THHTTHTH Penney’s game Walter Penney, Journal of Recreational Mathematics, October 1969, p.241

Probability of red sequence preceding blue HHHHHTHTHHTTTHHTHTTTHTTT HHH1/23/5 7/87/127/101/2 HHT1/21/3 3/43/81/23/10 HTH2/52/31/2 5/85/12 HTT2/52/31/2 1/41/8 THH1/81/41/2 2/32/5 THT5/125/81/2 2/32/5 TTH3/101/23/83/41/3 1/2 TTT1/27/107/127/83/5 1/2 n=3

Probability of red sequence preceding blue n=3

Which sequence is more likely to occur first n= Shorter wait always at least 50% probability of preceding longer one Average wait

Which sequence is more likely to occur first Average wait n=4

n=5n=6n=7n=8n=9 Which sequence is more likely to occur first

Average wait n=4

Which sequence has the shorter waiting time Average wait n=4

Which sequence is more likely to occur first Average wait n=4 Longer average wait is more likely to precede shorter one!

Shorter wait can precede longer wait Ignoring transposition of H,T, distinct cases are: HTHH (18) beats HHTT (16) with p=4/7 ≈ 0.57 HTHH (18) beats THHH (16) with p=7/12 ≈ 0.58 THHT (18) beats HHTT (16) with p=7/12 ≈ 0.58 THTH (20) beats HTHH (18) with p=9/14 ≈ 0.64 THTH has an almost 2/3 probability of preceding HTHH, yet takes longer to occur on average!

References Martin Gardner, Time Travel and other Mathematical Bewilderments, Same article, but also including proof based on average wait time: repository.net/il/user_contents/02/G Repository/repository/keidaironshu_063 _004_ pdfhttp:// repository.net/il/user_contents/02/G Repository/repository/keidaironshu_063 _004_ pdf Theorems X and Y in this article can be seen to be true based on the gambling approach shown in the following Martingale approach: Which sequence will occur first? Penney Ante: Counterintuitive Probabilities in Coin Tossing bact.mathcircles.org/files/Summer2010/PenneyAnte.pdf bact.mathcircles.org/files/Summer2010/PenneyAnte.pdf Penney’s game How to win at coin flipping flipping/ flipping/

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