Building a Contingency Table for On-time Delivery

Introduction

• Objective: Build a contingency table using probabilities of document delivery by two services.
• Example problem: Two copies of a document sent using Service A and Service B.

Given Probabilities

• Probability of on-time delivery with:
  • Service A: 0.9
  • Service B: 0.8
  • Both Services A and B: 0.75

Table Setup

• Rows: Events associated with Service A
  • Event A: Document arrives on time
  • Not A: Document does not arrive on time
  • Total probability of A: 0.9
• Columns: Events associated with Service B
  • Event B: Document arrives on time
  • Not B: Document does not arrive on time
  • Total probability of B: 0.8

Calculating Probabilities

1. Intersection of A and B:

   • Probability (A and B) = 0.75

2. Row A:

   • Total Probability = 0.9
   • Calculate missing cell: 0.9 - 0.75 = 0.15

3. Column B:

   • Total Probability = 0.8
   • Calculate missing cell: 0.8 - 0.75 = 0.05

4. Total Table Probability:

   • Must sum to 1.0
   • Accounted 0.8 for B, remaining: 1.0 - 0.8 = 0.2

5. Row Not A:

   • Total Probability = 0.1 (calculated: 1.0 - 0.9)
   • Fill remaining cells using arithmetic.

Verification

• Row Totals:
  • A: 0.75 + 0.15 = 0.9
  • Not A: 0.05 + 0.05 = 0.1
• Column Totals:
  • B: 0.75 + 0.05 = 0.8
  • Not B: 0.15 + 0.05 = 0.2
• Overall Check:
  • Sum of probabilities: 0.8 + 0.2 = 1.0
  • 0.9 + 0.1 = 1.0

Conclusion

• The contingency table accurately reflects the given probabilities and checks out through arithmetic and logical verifications.