In a city, 10% of people are vegetarians (V). Among vegetarians, 30% prefer biking as transport (B). Among non-vegetarians (Vc), 15% prefer biking. If a randomly selected person prefers biking, what is the probability that they are vegetarian? (Use Bayes' Theorem)

Question

BlackTom AI · Accepted Answer

I will break down the problem step by step and evaluate each option.
First, note the given priors and conditional probabilities: P(V) = 0.10, so P(Vc) = 0.90. Among vegetarians, P(B|V) = 0.30. Among non-vegetarians, P(B|Vc) = 0.15.
Now, compute the total probability of biking, P(B), using the law of total probability:
- Contribution from vegetarians: P(V) * P(B|V) = 0.10 * 0.30 = 0.03.
- Contribution from non-vegetarians: P(Vc) * P(B|Vc) = 0.90 * 0.15 = 0.135.
- Sum them for P(B): 0.03 + 0.135 = 0.165.
Next, apply Bayes’ theorem to find P(V|B):
P(V|B) = [P(B|V) * P(V)] / P(B) = (0.30 * 0.10) / 0.165 = 0.03 / 0.165 ≈ 0.1818.
Now assess each option:
- Option a: 0.1818. This matches the calculated value from Bayes’ theorem, so it could be a correct representation of P(V|B).
- Option b: 0.1. This would correspond to assuming P(V|B) equals the prior P(V) and ignores the information from B; it’s not consistent with Bayes’ calculation here.
- Option c: 0.175. This is close to 0.1818 but not equal; it would imply a slightly different arithmetic or rounding, but the exact Bayes result is 0.1818...
- Option d: 0.03. This value equals P(B and V) = P(V) * P(B|V), not the conditional probability P(V|B); it’s the joint probability for V and B, not the sought Bayesian posterior.
In summary, the correct approach yields a posterior probability around 0.1818, with the other options representing either the prior, a rounded approximation, or a joint probability instead of the conditional probability asked for.

In a city, 10% of people are vegetarians (V). Among vegetarians, 30% prefer biking as transport (B). Among non-vegetarians (Vc), 15% prefer biking. If a randomly selected person prefers biking, what is the probability that they are vegetarian? (Use Bayes' Theorem)单项选择题

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