Gambling on Intervals?

Calculus Level 5

Let \(\epsilon=\dfrac{1}{N}\).\[\] Choose a number at random between 0 and 1.
Choose a second number between \(\epsilon\) and \(1+\epsilon\).
Choose a third number between \(2\epsilon\) and \(1+2\epsilon\).
Continue in this way until you choose the \(N^{\text{th}}\) number between \(1-\epsilon\) and \(2-\epsilon\).\[\] If the probability that the first number you chose is the smallest is \(\bigg (\dfrac{a}{b\times N}\bigg )^c\), find \(\lfloor 100c\times(a+b) \rfloor\).\[\] Details and Assumptions:

  • Assume \(N\) is very large.

  • You might wish to make small approximations.

This is part of Ordered Disorder.

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