Expected quotes until your order fills

Question

Each time you post a resting order, it gets filled with probability 	frac{1}{8} before you cancel and re-post, independently each time. What is the expected number of postings until your first fill?

Accepted Answer

Let E be the expected number of postings until a fill. Each posting is an independent trial with fill probability p = 	frac{1}{8}. Condition on the first posting: - With probability 	frac{1}{8} it fills, done after 1 posting. - With probability 	frac{7}{8} it does not: one posting used, and independence resets us, expecting E more. E = 	frac{1}{8}(1) + 	frac{7}{8}(1 + E) \implies E = 1 + 	frac{7}{8}E \implies 	frac{1}{8}E = 1, so E = 8. Again this is the geometric mean 1/p = 8. Takeaway. Any "how many attempts until it works" question with a constant per-attempt success probability p a…

Expected quotes until your order fills

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