Probability that $5$ numbers chosen randomly in the interval $[0, 1]$ all lie in the middle half of the interval.

Your solution looks entirely correct to me. Strictly speaking, the phrase

$5$ numbers chosen randomly in the interval $[0,1]$

is a little vague, because it doesn't say if the picks are independent, but you can safely assume that, and then the probability of each event is $\frac12$, so the probability of them all is $\frac{1}{32}$. Well done!

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