Question

P.61 Of all spam messages, \(17.00 \%\) contain both the word "free" and the word "text" (or "txt"). For example, "Congrats!! You are selected to receive a free camera phone, \(t x t\) ****** to claim your prize." Of all non-spam messages, \(0.06 \%\) contain both the word "free" and the word "text" (or "txt"). Given that a message contains both the word "free" and the word "text" (or "txt"), what is the probability that it is spam?

Short answer

The probability that a message is spam, given that it contains both the words 'free' and 'text' or 'txt', is approximately 1 (or 100%).

Step-by-step solution

Defining the Variables

Let's define event A as the probability that a message is spam and event B as the probability that a message contains both 'free' and 'text' or 'txt'. We are given that P(B|A) = 0.17 and P(B|¬A) = 0.0006.

Understanding the Question

We are required to find P(A|B) i.e., the probability that a message is spam, given that it contains both the words 'free' and 'text' or 'txt'. We can use Bayes' theorem to solve this, which states that P(A|B) = [P(B|A) * P(A)] / P(B). However, we know P(B|A) and P(B|¬A) but P(A) and P(B) are not given directly.

Estimating P(A) and P(B)

To estimate P(A) and P(B), we can make an assumption since we don't know the exact values. We'll assume that the prior probability P(A), that a randomly given message is spam, is 0.5 (i.e., there is an equally likely chance that a randomly given message is spam or not spam). We can also estimate P(B) using the law of total probability. P(B) = P(B and A) + P(B and ¬A) = P(B|A) * P(A) + P(B|¬A) * P(¬A). Hence, P(B) = 0.17 * 0.5 + 0.0006 * 0.5 = 0.0853.

Calculating P(A|B)

By substituting P(A), P(B) and P(B|A) into the Bayes' theorem, we get P(A|B) = [P(B|A) * P(A)] / P(B) = (0.17 * 0.5) / 0.0853 ∼ 1. This implies that if a message contains both the words 'free' and 'txt', it is almost certainly a spam.

Key concepts

Bayes' Theorem

The essence of Bayes' Theorem is to revise probabilities given new evidence. It's a way to update our assumptions with actual data. Consider its application in real-life scenarios, such as email filtering. In our exercise, we've encountered spam detection. Here, Bayes' Theorem helps us determine the probability of an email being spam based upon the content of the message.

Formally, the theorem is expressed as \( P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} \). This formula might seem daunting at first, but each part has a clear meaning. \(P(A|B)\) is what we're after: the probability of our event A (receiving spam), given B (the message contains specific words). \(P(B|A)\) is the probability of observing event B given that A is true, while \(P(A)\) and \(P(B)\) are the probabilities of observing events A and B independently.

By incorporating the given data—such as the likelihood of 'free' and 'txt' appearing in spam and non-spam messages—we use Bayes' Theorem to update our prior belief about the likelihood of a message being spam based on its content. This adjustment based on evidence is what makes Bayes’ Theorem so powerful in statistical inference and decision-making.

Probability Estimation

Probability estimation involves determining the likelihood of an event occurring when not all information is known. In many real-world problems, such as the one presented in the exercise, we do not have all necessary probabilities readily available. Instead, we often rely on assumptions or prior knowledge to estimate these probabilities.

For example, the probability that a randomly selected message is spam—depicted as \(P(A)\) in our problem—was not provided. We made an assumption that there is an equal likelihood of a message being spam or not, setting \(P(A) = 0.5\). This kind of estimation is common practice, especially when precise data is unavailable or to create a simplified model.

Estimations can be refined with more information, emphasizing the importance of informed guesswork and the subsequent adjustment of such guesses as new data emerge. In the case of our spam filter, as we accumulate more information (such as overall spam rates or user-specific spam rates), we can adjust our probability estimates to be more accurate.

Law of Total Probability

The Law of Total Probability is a fundamental rule that relates marginal probabilities to conditional probabilities. It essentially breaks down the probability of a single event into the sum of probabilities of that event occurring in conjunction with several mutually exclusive scenarios.

In the context of our exercise, we use the Law of Total Probability to estimate \(P(B)\), the probability of a message containing the words 'free' and 'txt'. We consider two mutually exclusive scenarios: the message is either spam (\(A\)) or not spam (\(¬A\)). The law then states that \(P(B) = P(B|A) \times P(A) + P(B|¬A) \times P(¬A)\).

This calculation is crucial because P(B) represents the overall likelihood of any given message containing the specific words without regard to its spam status. By understanding how an event interacts with different contexts, we can better understand the overall likelihood of an event occurring. The law is particularly useful in complex systems where events can occur for various reasons, as with the numerous factors that might contribute to the presence of certain words in an email message.