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Real Numbers – Episode 12, Photographing the Chaotic Marching Band

Jane StreetSawyer
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  • Episode Topic: Application of expected value (EV) and the linearity of expectation to solve probabilistic problems.
  • Definition: Expected value represents the probabilistic average of a random process, calculated by summing outcomes weighted by their probabilities.
  • Key Property: Linearity of expectation states that $E[X + Y] = E[X] + E[Y]$ and $E[aX] = aE[X]$ for any random variables and constants, allowing the EV of complex sums to be calculated without determining specific outcome probabilities.

Review of Previous Episode (Haunted Forest Monster):

  • Scenario: A monster is equally likely to be Timid (1-6 damage), Aggressive (1-12), or Deadly (1-20).
  • First Attack EV:
    • Calculated directly as $41/6$ (approx. 6.83) by summing damage outcomes weighted by disjoint probabilities.
    • Calculated via linearity by summing the EV of each monster type weighted by their $1/3$ probability: $(1/3 \times 3.5) + (1/3 \times 6.5) + (1/3 \times 10.5) = 41/6$.
  • Second Attack EV (Conditional):
    • Initial Condition: First attack dealt 4 damage.
    • Bayesian Update: Updated probabilities for monster type given 4 damage:
      • Timid: $5/9$.
      • Aggressive: $5/18$.
      • Deadly: $1/6$.
    • Resulting EV: The updated expected damage for the second attack is $11/2$ (5.5), reflecting the higher likelihood of the monster being "Timid" after a low-damage roll.
    • Counterfactual: If the first attack was 16 damage, the monster would be known 100% to be Deadly.

Problem of the Week (Marching Band):

  • Scenario A (Independent Choices):
    • Parameters: 40 band members in 10 rows of 4; each member independently chooses Left or Right foot (50/50 probability).
    • Goal: Find the expected number of rows where all 4 members are in step.
  • Scenario B (Fixed Pool):
    • Parameters: Exactly 20 members start with Left foot and 20 with Right foot; members are randomly arranged into the 10 rows of 4.
    • Goal: Find the expected number of in-step rows under this constrained distribution.
  • Instruction: Solutions requiring work shown are due via the specified submission page.