# 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.

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