An occasionally dishonest casino uses two kinds of dice. A fair dice that has 1/6 probability of rolling any number, and a loaded dice that has 1/2 probability to roll a 3 and 1/10 probability to roll any other number. The probability that the casino switches from the fair to loaded dice is 0.01 and the probability of switching back from loaded to fair is 0.25.

Sequentially record dice by Q={q1, q2, q3...}, and numbers rolled by O={o1,o2,o3...}, for

example:q1=F, o1=2 means the first die is Fair, and by rolling the first die the number observed

is 2. We also know P(q1=F)=P(q1=L)=0.5.

Question:

Suppose we observe O={ o1=4,o2=2,o3=4,...,o1000=6} but do not observe Q={q1,q2,...q1000}, and after some calculation we know P(O,q1000=F)=a, P(O,q1000=L)=b, (0