To determine the probability that a document is spam given that it contains the word "llwws," we can use Bayes' Theorem. However, we need some additional information.
Let's define:
- \( P(\text{Spam}) \): The prior probability of a document being spam.
- \( P(\text{Not Spam}) \): The prior probability of a document not being spam.
- \( P(\text{llwws | Spam}) \): The probability of the word "llwws" appearing given that the document is spam.
- \( P(\text{llwws | Not Spam}) \): The probability of the word "llwws" appearing given that the document is not spam.
- \( P(\text{llwws}) \): The total probability of the word "llwws" appearing in a document.
From your data:
- You provided \( P(\text{llwws | Spam}) = 0.6 \).
- However, we do not have \( P(\text{Spam}) \) or \( P(\text{Not Spam}) \), nor \( P(\text{llwws | Not Spam}) \).
Assuming you meant to provide the percentage of the emails that are spam and not spam, let's denote:
- \( P(\text{Spam}) = p \)
- \( P(\text{Not Spam}) = 1 - p \)
- \( P(\text{llwws | Not Spam}) \): This is not provided, but if you have this information, we can proceed.
If we assume \( P(\text{llwws | Not Spam}) \) is significantly lower than \( P(\text{llwws | Spam}) \), we can proceed with a general approach.
Using Bayes' theorem:
\[ P(\text{Spam | llwws}) = \frac{P(\text{llwws | Spam}) \cdot P(\text{Spam})}{P(\text{llwws})} \]
For \( P(\text{llwws}) \):
\[ P(\text{llwws}) = P(\text{llwws | Spam}) \cdot P(\text{Spam}) + P(\text{llwws | Not Spam}) \cdot P(\text{Not Spam}) \]
If we could assume a value for \( P(\text{Not Spam}) \) or \( P(\text{llwws | Not Spam}) \), we could compute an answer.
Without these values, we cannot calculate an exact numeric answer. To find the probability that a document is spam given it has the word "llwws," you'll need to provide those additional probabilities.
Once you have:
- The prior probabilities \( P(\text{Spam}) \) and \( P(\text{Not Spam}) \)
- The probability \( P(\text{llwws | Not Spam}) \)
We can substitute those values into the formulas to solve for \( P(\text{Spam | llwws}) \).