A Markov chain has 3 possible states: A, B, and C. Every hour, it makes a transition to a different state. From state A, transitions to states B and C are equally likely. From state B, transitions to states A and C are equally likely. From state C, it always makes a transition to state A. (a) Write down the transition probability matrix. (b) If the initial distribution for states A, B, and C is P0 = ( 1 3 , 1 3 , 1 3 ), find the distribution of state after 2 transitions, i. e., the distribution of X2. (c) Show that this is a regular Markov Chain. (d) Find the steady-state distribution for this chain.

Martha has a deck of cards. she has lost some of the cards, and now the deck only contains nine spades, eleven diamonds, eight clubs, and twelve hearts. martha predicts that whenever she draws a card from the deck without looking, she will draw a club one-fifth of the time. which activity would best allow martha to test her prediction? a. randomly draw a card from the box and see if it is a club. b. randomly draw a card. then, continue to draw another card until all eight clubs are drawn. c. randomly draw and replace a card 120 times. then, observe how close to 30 times a club is drawn. d. randomly draw and replace a card 100 times. then, observe how close to 20 times a club is drawn.

The legs of a right triangle measure 6 meters and 8 meters. what is the length of the hypotonuse.

How many zeros does this polynomial function, y=(x-8)(x+3)^2