It is observed that 50% of mails are spam. There is a software that filters spam mail before reaching the inbox. Its accuracy for detecting a spam mail is 99% and chances of tagging a non-spam mail as spam mail is 5%. If a certain mail is tagged as spam find the probability that it is not a spam mail.

Question

Prepp Team · Accepted Answer

Spam Mail Probability Calculation Using Bayes' Theorem

This solution explains how to find the probability that a mail tagged as spam is actually not a spam mail, given specific accuracy rates for a spam filtering software.

Defining the Events

Let's define the events involved in this probability problem:

S: The event that a mail is actually Spam.
NS: The event that a mail is Not Spam.
T: The event that a mail is Tagged as Spam by the filter.

Given Probabilities

From the question, we are given the following probabilities:

The probability that a mail is spam: $ P(S) = 50\% = 0.50 $
The probability that a mail is not spam: $ P(NS) = 1 - P(S) = 1 - 0.50 = 0.50 $
The accuracy of the filter for detecting a spam mail (True Positive Rate): $ P(T|S) = 99\% = 0.99 $
The probability that the filter tags a non-spam mail as spam (False Positive Rate): $ P(T|NS) = 5\% = 0.05 $

Objective of the Question

We need to find the probability that a mail is not spam given that it has been tagged as spam. In notation, we need to calculate $ P(NS|T) $.

Applying Bayes' Theorem

Bayes' Theorem is the perfect tool for this problem. It helps us update our belief (probability) about an event based on new evidence. The formula is:

$ P(NS|T) = \frac{P(T|NS) \times P(NS)}{P(T)} $

To use this formula, we first need to calculate the total probability of a mail being tagged as spam, $ P(T) $.

Calculating the Total Probability of Tagging as Spam (P(T))

A mail can be tagged as spam in two ways:

It is actually spam and the filter correctly tags it as spam.
It is not spam, but the filter incorrectly tags it as spam.

The total probability $ P(T) $ is calculated as:

$ P(T) = P(T \cap S) + P(T \cap NS) $

Using the definition of conditional probability, $ P(A \cap B) = P(A|B) \times P(B) $, we can rewrite this as:

$ P(T) = P(T|S) \times P(S) + P(T|NS) \times P(NS) $

Now, let's plug in the values:

$ P(T) = (0.99 \times 0.50) + (0.05 \times 0.50) $ $ P(T) = 0.495 + 0.025 $ $ P(T) = 0.520 $

So, the overall probability that any given mail gets tagged as spam is 0.520.

Calculating the Final Probability P(NS|T)

Now we can substitute the values into Bayes' Theorem:

$ P(NS|T) = \frac{P(T|NS) \times P(NS)}{P(T)} $ $ P(NS|T) = \frac{0.05 \times 0.50}{0.520} $ $ P(NS|T) = \frac{0.025}{0.520} $

Simplifying the Result

To express the answer as a simple fraction, we can convert the decimals to fractions or multiply the numerator and denominator by 1000 to remove decimals:

$ P(NS|T) = \frac{0.025 \times 1000}{0.520 \times 1000} = \frac{25}{520} $

Now, we simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5:

$ P(NS|T) = \frac{25 \div 5}{520 \div 5} = \frac{5}{104} $

Conclusion

Therefore, if a certain mail is tagged as spam, the probability that it is actually not a spam mail is $ \frac{5}{104} $.

Prepp Team · Answer

3/108

Prepp Team · Answer

5/106

Prepp Team · Answer

5/48