If a thumbtack is tossed in the air, it can come to rest

Chapter , Problem 25

(choose chapter or problem)

If a thumbtack is tossed in the air, it can come to rest on the ground with either the point up or the point touching the ground. Find a thumbtack. Before doing any experiment, what do you think \(\pi\), the probability of it landing point up, is? Next, toss the thumbtack 20 times and graph the log likelihood of \(\pi\). Then do another experiment: Toss the thumbtack until it lands point up 5 times, and graph the log likelihood of \(\pi\) based on this experiment.

Find and graph the posterior distribution arising from a uniform prior on \(\pi\). Find the posterior mean and standard deviation and compare the posterior with a normal distribution with that mean and standard deviation. Finally, toss the thumbtack 20 more times and compare the posterior distribution based on all 40 tosses to that based on the first 20.