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5

The maximum $M_n$ of interest is such that $$(1-2/\sqrt{n})K_n\leqslant M_n\leqslant1,\quad\text{where}\quad K_n=\max\{X_i\mid 1\leqslant i\leqslant\sqrt{n}\}.$$ By independence, $P(K_n\leqslant x)=P(X_1\leqslant x)^{\sqrt{n}}=x^{\sqrt{n}}\to0$ when $n\to\infty$, for every $x$ in $(0,1)$. Hence $K_n\to1$ in probability, $(1-2/\sqrt{n})K_n\to1$ in ...

5

Knowing that the 6th coin flip is still 50-50 and completely independent from the last 5 should make the entire intro of your question void. You should ask - is it worth borrowing \$100 in order to have a 50-50 chance of winning an additional \$100. Everything else is just just noise. To answer this just think about the simple goal of maximising the ...

4


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