Coin toss: Probability of a run of certain length out of a longer sequence First, my probability game is pretty elementary and many of the probability formulas still look pretty intimidating.
That said, I am trying to understand how to calculate the probability of a run of $x$ within a larger set of total tosses.
Specifically, what is the probability of getting $7$ heads in a row in a total of $100$ tosses.
A formula that I am looking at, but not sure if it is correct, is as follows:
$\Pr = 1-(1-0.5^7)^{(100-7+1)} = .522$
Does the above formula accurately capture the likelihood of the event occurring?
 A: This question is a variant of the "hot hands" phenomenon, much studied wrt shot sequences in basketball and, more generally, "luck vs skill" questions. The standard assumption is that the calculation of the probability of a sequence of 7 heads in a row is an unconditional estimate across all 1,000 tosses. Using this approach, the conclusion is that the "hot hand" phenomenon does not exist, which has had significant importance for decision-making.
However, and as Andrew Gelman notes in a blog post from July 2015 titled Hey-guess what? There really is a hot hand! (1), this unconditional approach is incorrect. As documented in a paper referenced by Gelman (2), a better approach is a conditional estimate where, given a head, the probability of the next toss (or shot) also being a head initializes a new sequence of conditional estimates. Under this conditional method, the hot hand phenomenon is demonstrable.
Like you, I'm not an expert with probabilistic reasoning explained with Latex formulas. So I will leave the derivations and proofs to the experts writing the paper referenced by Gelman. It's quite readable.
(1) http://andrewgelman.com/2015/07/09/hey-guess-what-there-really-is-a-hot-hand/
(2) Miller and Sanjuro, Surprised by the Gambler's and Hot Hand Fallacies? A Truth in the Law of Small Numbers, https://papers.ssrn.com/sol3/papers2.cfm?abstract_id=2627354
