4. Probability
Bayes' Theorem
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According to data from a metro station, 28% of trains are delayed. When compared to weather data, it was found that 73% of train delays and 35% of on-time rides were on days with precipitation. Given there is precipitation, what is the probability the train will be delayed?
A
0.45
B
0.56
C
0.20
D
0.80
Verified step by step guidance1
Step 1: Define the events and probabilities. Let A represent the event that a train is delayed, and B represent the event that there is precipitation. From the problem, P(A) = 0.28 (28% of trains are delayed), P(B|A) = 0.73 (73% of delays occur on days with precipitation), and P(B|A') = 0.35 (35% of on-time rides occur on days with precipitation).
Step 2: Use the law of total probability to calculate P(B), the probability of precipitation. The formula is P(B) = P(B|A)P(A) + P(B|A')P(A'), where P(A') = 1 - P(A). Substitute the given values into the formula.
Step 3: Calculate P(A'), the probability that a train is on time. Since P(A) = 0.28, P(A') = 1 - 0.28 = 0.72.
Step 4: Substitute the values into the total probability formula: P(B) = (P(B|A) * P(A)) + (P(B|A') * P(A')). This will give you the overall probability of precipitation.
Step 5: Use Bayes' Theorem to find P(A|B), the probability that a train is delayed given precipitation. The formula is P(A|B) = (P(B|A) * P(A)) / P(B). Substitute the values for P(B|A), P(A), and P(B) into the formula to calculate the result.
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