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MMLU-Pro / 8066 / Assuming we are underground, and the only thing we can observe is whether a…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Assuming we are underground, and the only thing we can observe is whether a person brings an umbrella or not. The weather could be either rain or sunny. Assuming the P(rain)=0.6 and P(sunny)=0.4. Assuming the weather on day kk is dependent on the weather on day k−1k-1. We can write the transition probability as P(sunny ∣\mid sunny) = P(rain ∣\mid rain) = 0.7. The person has 60% chance to bring an umbrella when the weather is rain, and 40% chance to bring an umbrella when the weather is sunny, i.e. P(umbrella ∣\mid rain) = 0.6 and P(umbrella ∣\mid sunny) = 0.4. If we observe that the person (1) brought an umbrella on day 1, (2) did not bring an umbrella on day 2, (3) brought an umbrella on day 3. What is the probability that day 2 is raining?
Plain-text mathematical notation (without MathML)
Assuming we are underground, and the only thing we can observe is whether a person brings an umbrella or not. The weather could be either rain or sunny. Assuming the P(rain)=0.6 and P(sunny)=0.4. Assuming the weather on day k is dependent on the weather on day k−1. We can write the transition probability as P(sunny ∣ sunny) = P(rain ∣ rain) = 0.7. The person has 60% chance to bring an umbrella when the weather is rain, and 40% chance to bring an umbrella when the weather is sunny, i.e. P(umbrella ∣ rain) = 0.6 and P(umbrella ∣ sunny) = 0.4. If we observe that the person (1) brought an umbrella on day 1, (2) did not bring an umbrella on day 2, (3) brought an umbrella on day 3.  What is the probability that day 2 is raining?
Original LaTeX notation
Assuming we are underground, and the only thing we can observe is whether a person brings an umbrella or not. The weather could be either rain or sunny. Assuming the P(rain)=0.6 and P(sunny)=0.4. Assuming the weather on day $k$ is dependent on the weather on day $k-1$. We can write the transition probability as P(sunny $\mid$ sunny) = P(rain $\mid$ rain) = 0.7. The person has 60% chance to bring an umbrella when the weather is rain, and 40% chance to bring an umbrella when the weather is sunny, i.e. P(umbrella $\mid$ rain) = 0.6 and P(umbrella $\mid$ sunny) = 0.4. If we observe that the person (1) brought an umbrella on day 1, (2) did not bring an umbrella on day 2, (3) brought an umbrella on day 3.  What is the probability that day 2 is raining?

category

math

options

  1. 0.5167
  2. 0.25
  3. 0.2
  4. 0.7
  5. 0.6
  6. 0.3333
  7. 0.5
  8. 0.8
  9. 0.5833
  10. 0.4

src

theoremQA-Math

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Source and history

upstream_id / Original problem ID
8066
task_id
1a436d26-77ae-58d1-b948-3cb26b3f5b3b
task_key
default--test--8066
task_revision_id
1
config / split
default / test

Official source

initial import