 Maximum Likelihood With Unfair Dice Problems

Subject Mathematics Probability

Question

You toss a coin 20 times, coming up with the sequence (the spaces just help it look nice):
HHHTH HTHTH HHHTH THHTT
What’s the likelihood of it being a fair coin (prob of head = prob of tail = .5)?
Alternatively it could be an unfair coin with probability p of a head and 1-p of a tail.
What’s the likelihood as a function of p? What value of p maximizes the likelihood (or its ln)?

(5) Same sequence as in (5) except this time your friend gives it to you. You suspect him of
using two coins. If the last toss was a head he then uses a coin whose head-probability is p, if the
last toss was a tail he next uses a different coin with head-probability r. What is the likelihood
now as a function of p, r? What values of p, r maximize the ln likelihood?

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