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: Identify the given probabilities and assign them to appropriate events. Let A represent the event 'train is delayed' and B represent the event 'precipitation occurs'. From the problem, P(A) = 0.28 (28% of trains are delayed), P(A') = 1 - P(A) = 0.72 (72% of trains are on time), 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'). Substitute the known values into this formula: P(B) = (0.73)(0.28) + (0.35)(0.72).
Step 3: Use Bayes' Theorem to find the probability of a train being delayed given precipitation, P(A|B). The formula for Bayes' Theorem is P(A|B) = [P(B|A)P(A)] / P(B). Substitute the known values into this formula: P(A|B) = [P(B|A)P(A)] / [(P(B|A)P(A) + P(B|A')P(A'))].
Step 4: Simplify the numerator of the Bayes' Theorem formula, which is P(B|A)P(A). This represents the joint probability of both precipitation and a train delay. Similarly, simplify the denominator, which is the total probability of precipitation, P(B).
Step 5: Divide the simplified numerator by the simplified denominator to calculate P(A|B). This will give you the probability that a train is delayed given that there is precipitation.
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