Question

Can you help me with the following problem markov chains, explaining the solution step by step
An electromagnet may (or may not) change its polarity once per second. The probability that it changes is, respectively, 0.2 or 0.7 depending on whether the current state is +1 or -1.
a. Define states and transition matrix
b. Suppose the current state is 1, find the probability that it will be again after 4 seconds.
c. Within 24 hours. What is the probability that the state is -1?

Answer 1

Solla! States are There where and I two states namely +1 represent the electromagnet +1 and -1 bolarity . Transition matrix Given i to -1 0.2 Probability of going from from state + 1 onee per second Probability from state tl to + | ber once second I-0.2 0.8. to tl once Given, probabity ber probabity from staté -| second 0.7 - to -L once probability thom state per second 1-0.7 0.3 Thus the transition matrix denoted by P is from +1 To to P 11 0.7 +1 0.2

Solb)! let xo denodes the initial state vector, Probability that current state is to Probability that current state is 1 Since current state is given to be tl Xo : ] Then the the +1 state finst probability aftes entry of pt represent that it 4 seconds wil be again Pxo 1 0.8 0.2 0.7 0.3 O p²% P (Px. ) 0.8 0.8 0.7 0.3 0.2 0.2 0.78 0.22 p²x Р (Р°, 2 0.8 0.7 0.3 D-78 0.22 0.2 11 0.778 0.222

-0.2 0 Now perform R. LR 0.2 1 -7/2 Thus 11 q 7.44 Now as q, + 2 = 1 7 V qr dlo q and Z 9 of 219 state From the result vector within a represents the 24 hours. the state is a Thus the probability that within 24 is hours 2 १