A coin is flipped repeatedly. One man bets that the sequence TT will come up first, the other bets on HT instead. Is there a difference in odds?
2007-07-15 08:07:21 · 2 answers · asked by Scythian1950 7 in Science & Mathematics ➔ Mathematics
Nice problem. There is a difference. Just examine the sample spaces for small numbers of coins. Say Ted bets TT, Hattie bets HT.
HH
HT Hattie wins
TH
TT Ted wins
In two flips, each has a 1/4 chance of winning.
HHH
HHT Hattie wins
HTH (Hattie already won)
HTT (Hattie already won)
THH
THT Hattie wins
TTH (Ted already won)
TTT (Ted already won)
With three coins, Hattie gains a 2/8 probability of winning, while there is no new chance of a win for Ted (he's already won with TTT, and HTT does no good). So at this point, Hattie has a 1/2 probability of winning, Ted still 1/4.
With four coins, Hattie has new wins on HHHT and THHT, giving probability of 5/8 now, while Ted is still at 1/4.
In fact, Ted's probability of winning will never increase. If he doesn't win in two tosses with TT, that means there's an H in the sequence. Before there can be another TT, there has to be an HT, and Hattie will win.
Just to be thorough, with n tosses (n >= 2), only H...H and TH...H are tosses where no one has won. P(Ted wins) = 1/4, P(Hattie wins) = 3/4 - 2/2^n, leaving probability 2/2^n of needing to flip again.
2007-07-15 09:17:50 · answer #1 · answered by brashion 5 · 2⤊ 0⤋
no difference. independent events.
2007-07-15 15:11:38 · answer #2 · answered by hazel b grand 2 · 0⤊ 1⤋