Question

A person uses his car 30% of the time, walks 30% of the time and rides the bus 40% of the time as he goes to work, he is late 10% of the time when he walks. He is late 3% of the time when he drives, and he is late 7% of the time he takes the bus. Use Bayes' Theorem to find the probability he took the bus if he was late.

A. 0.418
B. 0.289
C. 0.853
D. 0.627
E. 0.455

Answer 1

Using Bayes' Theorem, the probability that the person took the bus given he was late is approximately 0.418, which is option A.

Step-by-step explanation:

To find the probability that the person took the bus given he was late, we can apply Bayes' Theorem. Let's use the given percentages to denote the probabilities:

• P(Car) = 0.30
• P(Walk) = 0.30
• P(Bus) = 0.40
• P(Late|Car) = 0.03
• P(Late|Walk) = 0.10
• P(Late|Bus) = 0.07

We are looking for P(Bus|Late), the probability that he was on the bus given that he was late. Using Bayes' Theorem, this is calculated as:

P(Bus|Late) = (P(Late|Bus) * P(Bus)) / (P(Late|Bus) * P(Bus) + P(Late|Car) * P(Car) + P(Late|Walk) * P(Walk))

Plugging in the numbers:

P(Bus|Late) = (0.07 * 0.40) / ((0.07 * 0.40) + (0.03 * 0.30) + (0.10 * 0.30))

P(Bus|Late) = 0.028 / (0.028 + 0.009 + 0.030)

P(Bus|Late) = 0.028 / 0.067

P(Bus|Late) = 0.4179

The probability that he took the bus given he was late is approximately 0.418, which corresponds to option A.