ANSWERS: 1

You've described the only two patterns (sequences) of six consecutive outcomes that are successful. All other sequences of length 6 are unsuccessful. How many possible sequences are there all together? 2 outcomes on the 1st toss (heads or tails); AND 2 outcomes 2nd toss; AND... The total number of possible outcomes on six consecutive tosses is 2^6 = 64. Each sequence has a 1/64th probability. But 2 out of 64 are successful. So the probability of success in any sequence of SIX CONSECUTIVE ROLLS is 2/64 = 1/32 = 0.03125 But in 100 throws there are 100/6 = 16.66 sequences of length 6, making success that many times more likely. Note that the six consecutives throws don't necessarily begin on the 1st, 7th, 13th etc., but on average that's the degree of redundancy of 100 throws instead of just six throws. This makes the answer (0.03125)*(16.667) = 0.52083 (52% likelihood). Now I don't feel fully confident in the correctness of this answer, because the preceding paragraph is not a rigorous method but rather a crude estimate. I suspect you'd have to more carefully enumerate the possible outcomes (from among 2^100) to get the exact answer. I might even be off by a factor of 2 or more. This is a toughie. Or am I missing something simple?
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