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Markov Chain

Markov Chain. Andrew Wang. Yum. 0.2. Probability. 0.8. 0.7. 0.3. Monte Carlo Simulation. Examples: What are the most commonly visited spots in the game of monopoly? Drunkard's Walk. Markov Chain. Each day, you choose to eat either grapes, cheese, or lettuce:

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Markov Chain

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  1. Markov Chain Andrew Wang

  2. Yum 0.2 Probability 0.8 0.7 0.3

  3. Monte Carlo Simulation Examples: What are the most commonly visited spots in the game of monopoly? Drunkard's Walk

  4. Markov Chain Each day, you choose to eat either grapes, cheese, or lettuce: • Choice today affects preferences tomorrow • cheese => tomato (0.5) || lettuce (0.5) • grapes => grapes (0.1) || cheese (0.4) || lettuce (0.5) • lettuce => grapes (0.4) || cheese (0.6) nom

  5. Terminology Absorbing Expected Number Probability Linear System

  6. How is this even related to CS?

  7. Gaussian Elimination GaussianElimination[m_?MatrixQ, v_?VectorQ] :=Last /@ RowReduce[Flatten /@ Transpose[{m, v}]] 2x + y - z = 8 -3x - y + 2z = -11 -2x + y + 2z = -3 [ 2 1 -1 | 8 ] [ 1 (1/3) (-2/3) | (11/3) ] [ -3 -1 2 | -11 ] ===> [ 0 1 2/5 | (13/5) ] [ -2 1 2 | -3 ] [ 0 0 1 | -1 ]

  8. Terminology Row Echelon Form Reduced Row Echelon Form Gaussian Elimination what on Earth?

  9. Example problems Flip a coin a bunch of times: Expected number of flips before getting 6 heads in a row? Roll a 6 sided die a bunch of times: Expected number of rolls before getting 6 consecutive identical values in a row?

  10. POTW (medium) 30 points N (N<1000, N even) players sit around a table; the game begins with two opposite players having one die each. On each turn, the two players with dice roll them. If a player rolls a 1, he passes the die to his neighbour on the left; if he rolls a 6, he passes the die to his neighbour on the right; otherwise, he keeps the die for the next turn. The game ends when one player has both dice after they have been rolled and passed; that player has then lost. What is the expected number of turns the game lasts? Give your answer rounded to ten significant digits.

  11. POTW (medium) 30 points N (N<1000, N even) players sit in circle. Players at opposite sides start with 6-side die Players roll the dice at the same time: • 1 => pass die to the left • 6 => pass die to the right • 2,3,4,5 don't mean anything Find the expected number of times until 1 player has both dice

  12. POTW (hard) 50 points An infinitely long random string of digits: p1p2p3p4p5p6p7p8p9p10p11p12p13p14p15... Every integer will occur as a substring in X at some index Q Ex: X = 12501672... Q2501= 2 , Q67 = 6 Given integer N ( N has <= 18 digits ), Find the expected value of QN Hint: use Knuth-Morris-Pratt Pattern Matching

  13. POTW (hard) 50 points, hints It can be proven that the expected value is always integer (doesn't mean I know how) For large N: built-in double is not precise enough Use high precision decimals (BigDecimal) or integers during gaussian elimination Hint: Use gaussian elimination for small N then find pattern.

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