Summary of Key Probability and Statistics Concepts

Probability Concepts

• Probability of Events

  • $P(A) = 0.45$ (shutdown > 30 days)

  • $P(B) = 0.35$ (budget includes tax credits)

  • $P(A \cap B) = 0.27$ (shutdown > 30 days and tax credits included)

• Probability Calculations

  • a. $P(A \cap B') = P(A) - P(A \cap B) = 0.45 - 0.27 = 0.18$

  • b. $P(A' \cap B') = 1 - P(A \cup B) = 1 - (P(A) + P(B) - P(A \cap B)) = 1 - 0.53 = 0.47$

  • c. $P(B | A) = \frac{P(A \cap B)}{P(A)} = \frac{0.27}{0.45} = 0.60$

Student Survey Data

• Economics Major and GPA Summary

  • Total students surveyed: 180

  • Low GPA (Econ): $P(Econ \cap Low) = \frac{36}{180} = 0.20$

  • Probability of being an Economics major: $P(Econ) = \frac{80}{180} = 0.44$

  • Given Econ major, probability of low GPA: $P(Low | Econ) = \frac{36}{80} = 0.45$

Turnitin Flagging Probabilities

• Flagging Calculations

  • c. $P(Human \cap Flagged) = P(Human) \times P(Flagged | Human) = 0.20 \times 0.35 = 0.07$

  • d. $P(Flagged) = P(AI \cap Flagged) + P(Human \cap Flagged) = 0.68 + 0.07 = 0.75$

  • e. $P(AI | Flagged) = \frac{P(AI \cap Flagged)}{P(Flagged)} = \frac{0.68}{0.75} \approx 0.91$

Conditional Independence

• Independence Check

  • $P(Econ \cap Low) = 0.20$ vs. $P(Econ) \times P(Low) = 0.4444 \times 0.5889 \approx 0.262$

  • Conclusion: Low GPA and majoring in Economics are NOT independent.

AI Detection Tool Analysis

• Probabilities from Tree Diagram

  • a. Missing values calculated from conditional probabilities

  • b. $P(AI \cap Flagged) = P(AI) \times P(Flagged | AI) = 0.80 \times 0.85 = 0.68$

Policy Suggestion

• AI Policy Recommendation

  • Suggest replacing automatic detection with a transparent, learning-centered policy to encourage student learning.