# The Gambler's Friend

Level pending

If one die is chosen from a bag of six dice each given a number from $$1$$ to $$6$$ such that the $$n^{th}$$ die has a $$\frac{2}{7}$$ chance of rolling $$n$$ and a $$\frac{1}{7}$$ chance of rolling any other of the numbers from $$1$$ to $$6$$. Given that the die chosen rolled the sequence $$(1,1,2,3,4,5,6)$$ $$k$$ times and the probability that it is the first die is $$\frac{1}{2}$$ or greater what is the minimum value of $$k$$

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