A six-sided die has an unknown number of faces marked with a six. Let k be this unknown number, which we would like to estimate. Our prior distribution for k is P(k = j) = (5/8, j = 1 |1/16, j= 0,2,3,4,5,6. When the die is thrown each face has an equal chance of showing. The observed data is that the die was thrown twice, and it showed a six exactly once. (a) Write down the likelihood for the observed data. What is the maximum likelihood estimate for k? (b) Derive the normalized posterior distribution for k. What is the posterior mean for k? (c) Find the posterior predictive probability that if the die is thrown again, it will not show six.

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