COSC 681  FALL 2016   HW 12/5/2016

Given a 2-headed coin, you toss the coin twice and get toss1 = H, toss2 = H  (Head, Head).

You don't know whether the coin is a fair coin (which has observable states
50% head, 50% tail) 
or a loaded (unfair) coin (with observable states 75% head, 25% tail).  

An enemy can swap the coin for a different one between tosses.

The Hidden Markov Model for the coin is located at this URL: 
https://webcourse.cs.technion.ac.il/236522/Spring2008/ho/WCFiles/tutorial05.ppt
(slide # 14)

The relevant probabilities are copied here:
Transition probabilities (hidden states)  p( Z(n) | Z(n-1) )
    
       |     fair        loaded
====================================
fair   |     0.9          0.1

loaded |     0.1          0.5




Emission probabilities  p( toss(k) | Z(k) )
  
  Z(k)        |        toss(k)         p( toss(k) | Z(k)
hidden state  |    observed state    probability
================================================
fair          |       H (heads)        0.5
fair          |       T (tails)        0.5
loaded        |       H                0.75
loaded        |       T                0.25



A.  What is the probability of tosses: H, H, given that a fair coin is being
used (note -- use HMM to answer this question. HOWEVER, We know from regular
probability the answer = 0.25)


B.  What is the probability of tosses: H, H, given that a loaded coin is being
used (same note as for A)


C. You throw H, H.  What is the probability the coin was fair on the first
toss and loaded on the second toss.

Hint:  use the joint probability formula:
p(toss1 = H, toss2 = H, Z1 = fair, Z2 = loaded) =
   p(Z1 = fair) * p (toss1 = H  | Z1 = fair) *
   p(Z2 = loaded | Z1 = fair) * p(toss2 = H | Z2 = loaded)