Limiting chance the max beats 1 minus 3/n

Question

Draw n independent numbers uniformly from [0,1] and let Mn be the largest. As n grows large, what is the limiting probability that Mn exceeds 1 - 	frac3n?

Accepted Answer

Work with the complement. The maximum is below t when all n draws are below t, so P(Mn \le t) = t^n. With t = 1 - \frac3n: P\!\left(Mn \le 1 - 	frac3night) = \left(1 - 	frac3night)^n \longrightarrow e^{-3}. Therefore P\!\left(Mn 1 - 	frac3night) \longrightarrow 1 - e^{-3} \approx 0.9502. Even a ceiling only \frac3n below the top is cleared by the maximum about 95\% of the time for large n. Takeaway. For the exceedance event, take 1 - e^{-c}: the maximum almost always pokes above a ceiling a few \frac1n units below 1.

Limiting chance the max beats 1 minus 3/n

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