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...}, forexample: q1=F, o1=2 means the first die is Fair, and by rolling the first die the number observedis 2. We also know P(q1=F)=P(q1=L)=0.5.

Question:

Suppose, you generate O={o1,o2,...,o200} as

> O

[1] 3 1 5 6 1 2 6 2 6 3 1 2 4 6 6 5 4 1 1 2 5 3 1 1 5 2 3 4 3 3 3 1 1 4 4 5 4

[38] 4 4 4 5 5 5 2 4 5 3 2 6 3 1 5 6 4 1 1 4 3 1 6 3 4 2 3 1 4 4 4 5 5 4 4 6 5

[75] 3 4 4 5 3 3 1 3 3 4 1 5 2 3 2 3 4 5 4 6 4 2 2 4 6 6 6 2 3 3 2 1 5 4 6 5 4

[112] 2 5 1 2 6 2 6 5 1 2 5 5 3 5 3 2 1 4 6 1 4 1 3 4 1 5 2 5 3 1 6 3 6 3 3 2 6

[149] 1 6 3 3 6 4 3 6 1 1 6 2 1 5 2 6 1 6 1 6 6 5 3 5 1 2 2 4 4 5 4 4 3 5 3 1 2

[186] 2 3 2 2 2 5 6 4 4 2 4 5 5 3 4

and Q={q1,q2,...,q200} as

> Q

[1] "L" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F"

[19] "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F"

[37] "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F"

[55] "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F"

[73] "F" "F" "F" "F" "F" "F" "L" "L" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F"

[91] "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "L" "L" "L" "L" "F" "F"

[109] "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F"

[127] "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "L" "L" "L" "L" "L"

[145] "L" "L" "L" "L" "L" "L" "L" "L" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F"

[163] "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F" "F"

Pretend you don't know Q, use Forward-Backward algorithm to estimate (Q'), and compare Q'with Q, what's the accuracy?print Forward table and Backward table, What are P(qi=F|O) and P(qi=L|O) for each i=1,2,...200 ? what is P(O) ? find P(q47=F|o1,o2...,o47), P(q47=F|o47,o48..,o200) , P(q56=L|o45,o46,...o60), P(o201=3|O)