To find the probability that a document is spam given that it contains the word "llwws", we can use Bayes' Theorem. However, we need a little more information about the overall prevalence of spam and not spam documents in your dataset.
From what you've provided:
- The probability that the word "aabbdd" is spam is 0.61.
- The probability that "llwws" is not spam is 0.6.
We can denote:
- \( P(\text{spam} | \text{llwws}) \): the probability that the document is spam given the presence of the word "llwws".
- \( P(\text{not spam} | \text{llwws}) = 0.6 \): the probability that the document is not spam given the presence of the word "llwws".
Let’s also introduce some notations:
- \( P(\text{llwws}) \): the total probability of the word "llwws" appearing in any document (spam + not spam).
- \( P(\text{spam}) \): the prior probability of a document being spam.
- \( P(\text{not spam}) = 1 - P(\text{spam}) \): the prior probability of being not spam.
Using the provided information and assuming we have the following relationships:
- If "llwws" has a probability of being not spam as 0.6, then: \[ P(\text{spam} | \text{llwws}) + P(\text{not spam} | \text{llwws}) = 1, \] thus, \[ P(\text{spam} | \text{llwws}) = 1 - 0.6 = 0.4. \]
With this information:
- The probability that a document is spam given that it contains "llwws" is 0.4, or 40%.
This leads us to state that if the word "llwws" is in a document, there is a 40% probability that it is spam.