Checking My Understanding of the Naive Bayes Theorem

[jisbon]

Homework Statement

Given the following statistics from a specific location:
60% of rainy days start out cloudy in the morning.
40% of all mornings are cloudy.
In December, it rains on average 9 out of 30 days.

In Dec, when it is a cloudy morning, what is the probability based on the given statistics that it is going to rain?

Relevant Equations

P(A|B) = P(B|A) P(A) / P(B)

I would like to check my understanding here to see if it is correct as I am currently stuck at the moment.
From the question, I can gather that:
P(Rain | Dec) = 9/30
P(Cloudy | Rain) = 0.6?
P(Cloudy | Rain) = 0.4

To answer the question:
P(Rain | <Cloudy, Morning, December> ) = P(Rain) * P(Cloudy|Rain) * P(Morning|Rain) * P(Dec|Rain)
= ? * 0.6 * ? * 9/30

This is going towards the Naive Bayes Theorem though, right? I think my initial thought process may be already wrong. Any guidance is greatly appreciated. Thank you!

 

[PeroK]

First, we have to assume that the data applies to December. If the data about cloudy and non-cloudy days is different in December from the annual average, then we have too little data to go on.

In that sense, it is just a straight calculation for December.

Homework Statement:: Given the following statistics from a specific location:
60% of rainy days start out cloudy in the morning.
40% of all mornings are cloudy.
In December, it rains on average 9 out of 30 days.

In Dec, when it is a cloudy morning, what is the probability based on the given statistics that it is going to rain?
Relevant Equations:: P(A|B) = P(B|A) P(A) / P(B)

I may have advised this before, but if you use the raw equations, then you risk getting lost in a sea of conditional probabilities. You should try a probability tree for these problems.

I would like to check my understanding here to see if it is correct as I am currently stuck at the moment.
From the question, I can gather that:
P(Rain | Dec) = 9/30

This is just the probability of rain for the problem. You don't need the conditional probability for December, as it is assumed across all calculations that we are in December.

P(Cloudy | Rain) = 0.6?

Correct. That's what "60% of rainy days start out cloudy means".

P(Cloudy | Rain) = 0.4

No. $0.4$ is the probability that any given day is cloudy in the morning.

To answer the question:
P(Rain | <Cloudy, Morning, December> ) = P(Rain) * P(Cloudy|Rain) * P(Morning|Rain) * P(Dec|Rain)
= ? * 0.6 * ? * 9/30

You're just lost now trying to use these horrible equations, rather than a nice probability tree!