You are given a coin and a thumbtack and you perform the following experiment: toss both the thumbtack and the coin 100 times. You get 60 heads and 40 tails for the coin, 70 heads and 30 tails for the...

Extracted text: You are given a coin and a thumbtack and you perform the following experiment: toss both the thumbtack and the coin 100 times. You get 60 heads and 40 tails for the coin, 70 heads and 30 tails for the thumbtack. You put Beta priors Beta(1,1), Beta(40, 60), Beta(30, 70), Beta(100, 100), Beta(1000, 1000), and Beta(100,000, 100,000) on the coin and thumbtack, respectively. (For both the coin toss and thumbtack toss experiments, you put these three priors, respectively.) Derive the MLE and MAP estimates for the coin and the thumbtack.