If an email filtering system has a 5% chance of falsely marking a legitimate email as spam and a 1% chance of letting a spam email through, what is the probability of a flagged email being legitimate if 10% of incoming emails are spam?

Official Answer

Certainly, approaching a question of this nature, especially in the context of a Data Scientist role, allows me to draw upon my rich experiences in statistical modeling, data analysis, and my understanding of Bayesian probability. So, let's break down the problem step-by-step to ensure we're on the same page.

First, let's identify the key pieces of information provided: - The probability of falsely marking a legitimate email as spam is 5%. - The probability of letting a spam email through is 1%, though for this specific question, this information is supplementary as we're focusing on the flagged emails. - 10% of incoming emails are spam, which inherently means 90% are legitimate.

Our goal is to find the probability of a flagged email being legitimate. To solve this, we can apply Bayes' theorem, which in the context of this problem, allows us to update our prior beliefs (in this case, the initial classification of emails as spam or legitimate) based on new evidence (the email being flagged).

Taking the given probabilities, let's denote: - (P(Spam) = 0.10) as the probability of any email being spam. - (P(Legitimate) = 0.90) as the probability of any email being legitimate. - (P(Flagged|Legitimate) = 0.05) as the probability that a legitimate email is flagged as spam. - (P(Flagged|Spam) = 0.99) as the probability that a spam email is flagged, considering the inverse of letting a spam email through.

The question asks for (P(Legitimate|Flagged)), the probability that an email is legitimate given that it has been flagged. To compute this, we apply Bayes' theorem:

[ P(Legitimate|Flagged) = \frac{P(Flagged|Legitimate) \cdot P(Legitimate)}{P(Flagged)} ]

Where (P(Flagged)) is the total probability of an email being flagged, which can be calculated as:

[ P(Flagged) = P(Flagged|Legitimate) \cdot P(Legitimate) + P(Flagged|Spam) \cdot P(Spam) ]

Plugging in the numbers:

[ P(Flagged) = 0.05 \cdot 0.90 + 0.99 \cdot 0.10 = 0.144 ]

Therefore, the probability of a flagged email being legitimate is:

[ P(Legitimate|Flagged) = \frac{0.05 \cdot 0.90}{0.144} \approx 0.3125 ]

So, in conclusion, when approached with a flagged email, there's approximately a 31.25% chance that it's actually a legitimate email. This insight, derived from a probabilistic framework, is not only crucial for understanding the performance and potential pitfalls of our email filtering system but also showcases the analytical rigor I bring to solving complex, data-driven problems. This approach is emblematic of how I tackle challenges, leveraging statistical principles to extract actionable insights, a skill set I'm eager to bring to your team to drive data-informed decisions.