Markov Chains and the Memoryless Assumption · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Tomorrow depends only on today

Mathematics · Probability · ages 22-24
Name ______________________   Date ____________

  1. Sunny is followed by Sunny with probability 0.8 and Cloudy with 0.2. Cloudy is followed by Sunny with 0.4 and Cloudy with 0.6. Rows are today, columns are tomorrow. Which matrix fits?

    • Rows (0.8, 0.4) and (0.2, 0.6)
    • Rows (0.8, 0.2) and (0.4, 0.6)
    • Rows (0.2, 0.8) and (0.6, 0.4)

  2. Why must each row of a transition matrix sum to 1?

    • Because tomorrow always brings some state
    • Because every chain has two states
    • Because columns must sum to 0

  3. In a Markov chain, the chance of tomorrow's state can depend on the state from three days ago.

    Circle one:   True   False

  4. Sunny is followed by Sunny with probability 0.8 and Cloudy with 0.2. Cloudy is followed by Sunny with 0.4 and Cloudy with 0.6. Rows are today, columns are tomorrow. Which matrix fits?

    • [[0.8, 0.2], [0.4, 0.6]]
    • [[0.8, 0.4], [0.2, 0.6]]
    • [[0.2, 0.8], [0.6, 0.4]]
    • [[0.8, 0.2], [0.6, 0.4]]

  5. Today is Sunny. From Sunny, tomorrow is Sunny with probability 0.8 and Cloudy with 0.2; from Cloudy, Sunny with 0.4 and Cloudy with 0.6. What is the probability of Cloudy two days from now?

    Answer: ______________

  6. Today is Sunny. From Sunny, tomorrow is Sunny with probability 0.8 and Cloudy with 0.2; from Cloudy, Sunny with 0.4 and Cloudy with 0.6. What is the probability of Sunny two days from now?

    Answer: ______________

  7. A forecast using the last two days is not a Markov chain, but tracking pairs of consecutive days as the state fixes that.

    Circle one:   True   False

  8. Sam says: a forecast using the last two days is not a Markov chain, but tracking pairs of consecutive days as the state fixes that. Is Sam right?

    Circle one:   True   False

  9. Today is Cloudy. From Sunny, tomorrow is Sunny with probability 0.8 and Cloudy with 0.2; from Cloudy, Sunny with 0.4 and Cloudy with 0.6. What is the probability of Cloudy two days from now?

    • 0.44
    • 0.36
    • 0.6
    • 0.52

  10. Today is Cloudy, with the same chances as above. What is the probability of Sunny two days from now?

    • 0.44
    • 0.72
    • 0.56
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Tomorrow depends only on today W1-mt_2wDa_JxJ_o-s1

  1. Rows (0.8, 0.2) and (0.4, 0.6) · Row Sunny must read 0.8 then 0.2, and row Cloudy 0.4 then 0.6.
  2. Because tomorrow always brings some state · From any today, the chances over all possible tomorrows exhaust everything.
  3. False · The memoryless assumption lets tomorrow depend only on today.
  4. [[0.8, 0.2], [0.4, 0.6]] · Rows are today, columns are tomorrow: row Sunny is 0.8 then 0.2, row Cloudy is 0.4 then 0.6.
  5. 0.28 · Two paths end at Cloudy: 0.8 times 0.2 plus 0.2 times 0.6, which is 0.28.
  6. 0.72 · Two paths lead back to Sunny: 0.8 times 0.8 plus 0.2 times 0.4, which is 0.72.
  7. True · Two-day memory breaks the rule, and pairs of days restore it.
  8. True · Two day memory breaks the Markov rule, and pairs of days restore it.
  9. 0.44 · From Cloudy: 0.4 times 0.2 plus 0.6 times 0.6 gives 0.44.
  10. 0.56 · From Cloudy: 0.4 times 0.8 plus 0.6 times 0.4 gives 0.56.
Worksheet · LightMySky