$\dfrac{13}{24}$
$\dfrac{7}{12}$
$\dfrac{2}{3}$
$\dfrac{1}{2}$
$\dfrac{5}{8}$

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This problem is a disguised version of a simple expected value problem. By the law of large numbers, which states that the arithmetic mean of a result approaches the expected value as more trials are performed, we can assume that maximizing our chances of winning means finding the expected value of winning. Thus, if we take the arithmetic mean of each range of numbers and then take the arithmetic mean of that, we should reach our answer.

Alice's expected value: $\frac{0 + 1}{2} = \frac{1}{2}$

Bob's expected value: $\frac{\frac{1}{2} + \frac{2}{3}}{2} = \frac{7}{12}$

Optimized expected value: $\frac{\frac{1}{2} + \frac{7}{12}}{2} = \boxed{\frac{13}{24}}$

If you're interested in further research:

Law of Large Numbers

Expected Value