BWAHAHAHAHAHAHAHAH
Yeah sparky, you go with that.
2^6 is not the statistical probability across 6 coin tosses.
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What actually happens is that when we examine the possibility of an unsuccessful run of heads at toss
i, we slightly bias the outcome at toss
i+1. A demonstration will show exactly how this happens.
For the sample problem of a sequence of two heads out of four tosses, we can first examine the chance of a negative outcome starting at toss 1. There are just four possible outcomes that have two tosses starting at position 1: HH HT TH TT And only one of these tosses yielded two heads in a row, so the probability of not seeing two heads after two tosses is 3/4.
But now when we look at the sequence of tosses starting at position two, we have to throw out the outcomes where we had two heads at toss one - we've already seen two heads, so we can't continue flipping coins in those outcomes. So our universe of possible outcomes is now a bit different: HTH HTT THH THT TTH TTT Instead of eight outcomes, we have six. And if we look at the first toss seen in position two, instead of having an even distribution of heads and tails, you can see that sample is biased: only two have a head in position two, while four have tails. So the chances of not seeing two heads starting at position two increases to 5/6. Note that this change in probability occurs because we have selected only those outcomes without a streak of two heads at position one.
Likewise, when we look at the possible outcomes for streaks starting at position three, we get a different probability again. Because we have to throw out one sequence in the previous test, the universe of possible outcomes is now limited to: HTHH HTHT HTTH HTTT THTH THTT TTHH TTHT TTTH TTTT So now we have just ten possible outcomes, and two of those will produce the desired outcome, meaning the probability has changed to 4/5.
So what is the probability of all three possible positions not containing a streak? That would be (3/4)*(5/6)*(4/5) which reduces nicely to 1/2, the correct answer.}
20 Heads In a Row - What Are the Odds?
Factor in the streak and it is actually 2^6^6 or 64^6