Question

coin 1 has probability 0.7 of coming up heads, and coin 2 has probability of 0.6 of coming up heads. we flip a coin each day. if the coin flipped today comes up head, then we select coin 1 to flip tomorrow, and if it comes up tail, then we select coin 2 to flip tomorrow. find the following:

a) the transition probability matrix P

b) in a long run, what percentage of the results are heads?

c) if the results on the first day is a head, what is the proability that the results on the second day and the third day are both tails?

Answer 1

a)

We need to set up a two state Markov Chain.

State 0 if coin 1 is flipped initially,

State 1 if coin 2 is flipped initially.

P00=.7,  P01=.3, P10=.6, P11=.4

Hence the Transition Matrix

b)

this will be P^n when n tend to

which can be solved using

=    0.6667    0.3333

hence

in a long run, what percentage of the results are heads

= 0.66667 = 2/3

c)

since head comes on first day

coin 1 will be tossed

hence probability of getting tail on second day = 0.3

and given tail on second day , probability of getting tail on third day = 0.4 {as coin 2 will be tossed}

hence required probability = 0.3*0.4 = 0.12