Bayes' Theorem Problem: Taxi Companies and Accidents
Problem Statement
- City: Two taxi companies, A and B.
- Company A: 60% of the taxis.
- Company B: 40% of the taxis.
Objective
Calculate the probability that a taxi involved in an accident belongs to Company B.
Approaches to Solve
Approach 1: Tree-Based Approach
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Assume 1000 Taxis in the City
- Company A: 600 taxis
- 3% involved in accidents → 18 taxis
- 97% not involved in accidents → 582 taxis
- Company B: 400 taxis
- 6% involved in accidents → 24 taxis
- 94% not involved in accidents → 376 taxis
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Total Taxis Involved in Accidents: 42 taxis (18 from A and 24 from B)
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Calculate Probability
- Probability (B | Accident) = Taxis from B involved in accidents / Total taxis involved in accidents
- Probability (B | Accident) = 24 / 42 = 0.57
Conclusion: Option B (0.57) is correct.
Approach 2: Formula-Based Approach (Bayes' Theorem)
Bayes' Theorem Formula
- Formula: P(A | B) = (P(B | A) * P(A)) / P(B)*
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Given Data
- P(A) = 0.6 (Probability of selecting a taxi from A)
- P(B) = 0.4 (Probability of selecting a taxi from B)
- P(Accident | A) = 0.03
- P(Accident | B) = 0.06
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Total Probability of an Accident
- P(Accident) = P(A) * P(Accident | A) + P(B) * P(Accident | B)
- P(Accident) = 0.6 * 0.03 + 0.4 * 0.06
- P(Accident) = 0.018 + 0.024 = 0.042
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Calculate Probability
- P(B | Accident) = (P(Accident | B) * P(B)) / P(Accident)
- P(B | Accident) = (0.06 * 0.4) / 0.042
- P(B | Accident) = 0.024 / 0.042 = 0.57
Conclusion: Probability that a taxi involved in an accident belongs to Company B is 0.57.
Optional: Validating with Tree Diagram
- Probability Values
- P(A): 0.6 (1000 * 0.6 = 600)
- P(B): 0.4 (1000 * 0.4 = 400)
- P(Accident | A): 0.03 (600 * 0.03 = 18)
- P(No Accident | A): 0.97 (600 * 0.97 = 582)
- P(Accident | B): 0.06 (400 * 0.06 = 24)
- P(No Accident | B): 0.94 (400 * 0.94 = 376)
Final Answer: The probability is 0.57.