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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