This question is from Quantguide
You're practicing soccer by taking
$100$ penalty kicks. Assume that you have made the first goal but missed the second. For each of the following kicks, the probability that you score is the fraction of goals you've made thus far. For example, if you made $17$ goals out of the first $30$ attempts, then the probability that you make the 31st goal is $\frac{17}{30}$. After 100 attempts, including the first two, what is the probability that you score exactly $66$ penalty kicks?
I tried solving , $\Pr(X_i = 1 | X_1 = 1 , X_2 = 0 \text{ and } i>2)$ if we have no information about past $i-3$ kicks is $0.5$ , but here we have some information like in total we want $66$ kicks so I think every configuration is not equally likely and the cases will be huge . I am unable to approach this question further . Any help will be great .
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user577215664
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Welcome to MSE! As you are a new contributor to the site. I suggest you read some basic information about writing mathematics at this site, see, e.g. How can I format mathematics, There should be universal LaTeX/MathJax guide for sites supporting it, and MathJax basic tutorial and quick reference – Afntu Aug 30 '24 at 13:05
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This is exactly the same setup as https://math.stackexchange.com/questions/4958393/binomial-random-variable-with-varying-probability-of-success/, so that should help. – Especially Lime Aug 30 '24 at 13:13
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This is essentially the classic Polya urn model, for which we have a tag pointing to similar questions. – Henry Sep 07 '24 at 23:35