# Use MLE to estimate the priors 𝑃𝑖(𝑗) for every pair of 𝑖,𝑗. (b). Sometime later, one of the dice disappeared. You (as the casino owner) need to find out which one.

INSTRUCTIONS TO CANDIDATES

In a casino, two differently loaded but identically looking dice are thrown in repeated runs. The frequencies of numbers (number of times each number has been observed) observed in 40 rounds of play are as follows:

Dice 1, [Nr, Frequency]: [1,5], [2,3], [3,10], [4,1], [5,10], [6,11] Dice 2, [Nr, Frequency]: [1,10], [2,11], [3,4], [4,10], [5,3], [6,2] (a) Denote by 𝑃𝑖(𝑗) the probability of getting the number 𝑗 (for every 𝑗 ∈ {1, ⋯ ,6}) using the dice i (for every i∈ {1,2})).

Use MLE to estimate the priors 𝑃𝑖(𝑗) for every pair of 𝑖,𝑗. (b). Sometime later, one of the dice disappeared. You (as the casino owner) need to find out which one.

The remaining one is now thrown 40 times and here are the observed counts: [1,8], [2,12], [3,6], [4,9], [5,4], [6,1]. Use a Bayes’ rule to decide the identity of the remaining dice.

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