Identification of spam emails and text message (so as to create a message-filter) can be accomplished using the Bayes’ rule. A database of SMS from cellphone users contain 774 spam messages out of a total of 5574 message i.e., 13.40% are marked as spam.
1 point
The term “free” is contained in 4.75% of all messages. 3.57% of all messages contain the word “free” and are marked as spam. Find the probability that a message contains the word “free”, given that it is spam.
A. 0.266
B. 0.752
C. 0.737
D. 0.978
1 point
What is the probability that a message is spam, given that it contains the word “free”?
A. 0.266
B. 0.752
C. 0.737
D. 0.978
1 point
7.01% of all messages contains the word “text” (or “txt”). The word “text” (or “txt”) is contained in 38.55% of all spam messages. Find the probability that a message is spam, given that it contains the word “text” (or “txt”).
A. 0.266
B. 0.752
C. 0.737
D. 0.978
1 point
Of all spam messages, 17.00% contain both the word “free” and the word “text” (or “txt”). 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?
A. 0.534
B. 0.752
C. 0.737
D. 0.978
1 point
Given that a message contains the word “free” but does NOT contain the word “text” (or “txt”), what is the probability that it is spam?
A. 0.534
B. 0.752
C. 0.737
D. 0.978
Answers
Given : A database of SMS from cellphone users contain 774 spam messages out of a total of 5574 message i.e., 13.40% are marked as spam.
The term “free” is contained in 4.75% of all messages. 3.57% of all messages contain the word “free” and are marked as spam.
7.01% of all messages contains the word “text” (or “txt”). The word “text” (or “txt”) is contained in 38.55% of all spam messages
To Find: the probability that a message contains the word “free”, given that it is spam.
probability that a message is spam, given that it contains the word “free”?
Solution:
Conditional probability formula
P(A|B) = P(A∩B)/P(B)
P(B|A) = P(A∩B)/P(A)
13.40% are marked as spam.
The term “free” is contained in 4.75% of all messages.
3.57% of all messages contain the word a “free” and are marked as spam.
probability that a message contains the word “free”, given that it is spam.
= P ( free and spam) / P( spam)
= 3.57% / 13.40%
= 0.266
probability that a message is spam, given that it contains the word “free
= P ( free and spam) / P( free)
= 3.57% / 4.75%
= 0.7516
= 0.752
7.01% of all messages contains the word “text” (or “txt”).
The word “text” (or “txt”) is contained in 38.55% of all spam messages
13.40% are marked as spam
Hence P ( spam message containing text) = 38.55% * 13.40%
the probability that a message is spam, given that it contains the word “text” (or “txt”).
= P ( spam message containing text/txt) / P (containing text/txt)
= 38.55% * 13.40% / 7.01%
= 0.7369
= 0.737
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