Real Numbers – Episode 13, How Many Snowboards Need Sharpening

Problem Statement

• Rachel provides private snowboarding lessons six days per week, with six available snowboards.
• Each lesson involves a student randomly selecting one of the six boards.
• At the week's end, every board used at least once must be sharpened.
• The objective is to calculate the expected number of snowboards requiring sharpening.

Mathematical Methodology: Linearity of Expectation

• The solution leverages indicator variables ($X_i$) to decompose a complex random variable ($X$) into simpler components.
• An indicator variable equals 1 if a specific condition is met and 0 otherwise.
• The expected value of an indicator variable is exactly the probability that the condition occurs ($E[X_i] = P(X_i=1)$).
• Linearity of expectation allows the total expected value to be calculated as the sum of individual expected values ($E[X] = \sum E[X_i]$), regardless of whether the variables are independent.

Case Study: Marching Band (Prior Episode)

• Scenario A (Independent Selection):
  • Context: 40 band members arranged in 10 rows of 4; each member starts with left or right foot with equal probability ($1/2$).
  • Condition for success: An "in-step" row occurs if all four members in a row start with the same foot.
  • Probability calculation: $P(\text{all left}) + P(\text{all right}) = (1/2)^4 + (1/2)^4 = 1/16 + 1/16 = 1/8$.
  • Result: Expected number of in-step rows is $10 \times (1/8) = 1.25$.
• Scenario B (Dependent Selection):
  • Context: Exactly 20 members start with the left foot and 20 with the right, arranged randomly.
  • Probability calculation: The probability the first row is in-step is derived from the reduced pool ($19/39 \times 18/38 \times 17/37$), resulting in a probability of $51/481 \approx 0.106$.
  • Result: Expected number of in-step rows is $10 \times (51/481) = 510/481 \approx 1.06$.
• Independence Analysis:
  • In Scenario B, indicator variables are not independent; knowing one row is in-step alters the probability for subsequent rows.
  • Despite this dependence, linearity of expectation remains valid for calculating the total sum.
  • The expected value decreases (1.25 to 1.06) because fixing the global distribution makes it harder to fill rows with a single foot type once partial matches occur.

Forward-Looking Statement

• The transcript concludes by inviting the audience to apply the indicator variable method discussed in the marching band problems to solve the snowboarding expected value problem.