Sorry if the title is confusing I couldn't think of a snappy way to describe this.
The raw probability of a getting at least one tails in 15 coin flips is 0.99997 (using the binomial distribution), but I am wondering if there has already been 6 successive heads does this change the probability of tails occurring?
Another way to frame my question is: if you were betting on tails coming up in the next nine coin flips, but you waited until six heads in a row had already appeared are your odds, of a tails appearing in the next nine coin flips, better than if you had simply bet on the first coin flip?
I don't quite now the proper terminology so I am sure this is confusing, please feel free to ask clarifying questions, and thank you a ton for reading this post
No, they are not better. In the first scenario, you are looking for the probability that you get at least $1$ tail in the first $9$ flips, which is just $\frac{511}{512}$ cause there is a $\frac{1}{512}$ chance of no tails.
However, in the second situation, if you already know that the first $6$ flips are head, you took the random element away from them, so they aren't included in your probability calculations. Thus, you only focus on the next $9$ flips, and thus there is a $\frac{511}{512}$ chance as calculated before.
Basically, the $9$ flips do not care about what happened before because each flip is independent.