Problem 16.69 (Simpson's Paradox). Simpsons paradox arises when you analyze a conditional probability by breaking it down into cases. At a famous university, renowned for science and engineering, an investigation into gender asymmetry in graduate admissions, produced this (approximate) data. Science/ Engineering Languages/Humanities # Applicants #Admitted # Applicants #Admitted Male 825 533 273 16 Female 108 81 352 34 For a randomly picked applicant: (a) Compute P[Admitted | Male] and P[Admitted | Female]. Is there a case for gender bias in admissions? (b) Now consider the disciplines separately and compute P Admitted | Male and Applied to Science/Engineering] P Admitted | Male and Applied to Languages/Humanities] P Admitted | Female and Applied to Science/ Engineering] P Admitted | Female and Applied to Languages/Humanities] Is there a case for gender bias in admissions? (The full and by-discipline conditional probabilities lead to opposite conclusions. Extreme care must be taken when approaching high-octane issues with conditional probabilities. Though our numbers are approximate, the conclusions match the real-life investigation. The critical issue that must be understood is why significantly fewer female students apply to science & engineeding.)