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The following observation is made over the two periods:

Note that

$A = \left \{ (u, d),\ (d, u),\ (u, u) \right \}$

and hence

$\begin{align*} Pr(at\: least\: one\: downswing) &=1-Pr(all\:upswing)\\ &=1-\frac{1}{16}\\ &=\frac{15}{16} \end{align*}$

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