First dont bother the downvote on this answer. The downvoter never even read this answer. With that said, I go on to my answer.
All the answers to this question which says 50% are WRONG. Correct Answer is 90%. After I explain my answer, I shall explain where each answer has made mistake.
First we must understand one simple concept, if it is said that given an event A has occurred, what is the probability of B occurring, then it is given by the expression: P (B | A)
Now I transform each line of the given question to above format.
1. "10% of all email you receive is spam" => Given a mail, probability that it is actually spam is 0.1
=> P (Actually Spam | Mail) = 0.1
2. "90% of the mails it marks as spam are indeed spam" => Given a mail which is marked as spam (by filter), probability that it is actually spam is 0.9
=> P ( Actually Spam | Marked Spam ) = 0.9
3. "90% of spam mails are correctly labeled as spam" => Given a mail which is actually spam, probability that it is marked spam (by filter) is 0.9
=> P ( Marked Spam | Actually Spam ) = 0.9
4. "see a mail marked spam by your filter, what is the probability that it really is spam" => Given a mail which is marked spam (by filter), probability that it is actually spam is what ?
=> P ( Actually Spam | Marked Spam ) = ?
Clearly, we can see that this is already given itself in the problem statement in Point 2. "90% of the mails it marks as spam are indeed spam"
∴ P ( Actually Spam | Marked Spam ) = 0.9 (Answer D) .
Now I am coming to the part of discussing where each question made mistake (please don't consider it as my arrogance :-)
1. by srestha Veteran
10% email are spam, i.e. 90% email are not spam
90% of mail marked as spam is spam, 10% mail marked as spam are not spam
By Bayes theorem the probability that a mail marked spam is really a spam
=Probability of being spam and being detected as spam / Probability of being detected as spam
Now,
Numerator = Probability of being actually spam and being marked as spam = P(Marked Spam | Actually Spam) * P(Actually Spam) = 0.9 * 0.1
Denominator = Probability of being marked as spam = P(Marked Spam) = P(Marked Spam | Actually Spam) * P(Actually Spam) + P(Marked Spam | Actually NOT Spam) * P(Actually NOT Spam)
The denominator basically uses the equation: P(A) = P(A ∩ B) + P (A ∩ Bc) = P(A | B) * P(B) + P(A | Bc) * P(Bc)
Until this portion, everything is correct. Now the mistake:
she considered P(Marked Spam | Actually NOT Spam) = 1 - P(Marked Spam | Actually Spam) = 1 - 0.9
which is basically a way of saying P(A | Bc) = 1 - P(A | B). (WHICH IS NOT CORRECT)
2. by Ayush Upadhyaya Loyal
In his diagram, in the top rightmost portion, he considered
Email is marked as Spam by filter = 0.1
i.e. He considered P(Marked Spam | Actually NOT Spam) = 0.1. This is a mistake.
He gave the reason:
This is because it is given 90% of the mails are correctly marked as spam, Means only 10% are incorrectly marked as spam)
This is also the same mistake as srestha Veteran's answer. Basically the mistake is that in the second statement he considered P(Marked Spam | Actually Not Spam) = 1 - P(Marked Spam | Actually Spam) which is not correct as explained previously.