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Chapter 11: Problem 60

The word "free" is contained in \(4.75 \%\) of all messages, and \(3.57 \%\) of all messages both contain the word "free" and are marked as spam. (a) What is the probability that a message contains the word "free", given that it is spam? (b) What is the probability that a message is spam, given that it contains the word "free"?

Short Answer

Expert verified
The probability that a message contains the word 'free', given that is spam is \(75\%\). And, the probability that a message is spam, given that it contains the word 'free' is also \(75\%\).

Step by step solution

01

Calculation of Probability for Part (a)

We know that the event 'a message contains the word 'free'' has occurred \(3.57 \%\) of the times and also the message is marked as spam. We also know that a message is marked as spam \(4.75 \%\) of the times. We're asked to compute \(P('free' | 'spam')\), which by definition of conditional probability, is \(P('free' 鈭� 'spam') / P('spam') = 0.0357 / 0.0475 = 0.75 = 75\% \).
02

Calculation of Probability for Part (b)

Now we need to find probability of a message marked as spam given it contains the word 'free'. With basic probability, this is given by \(P('spam' | 'free')\), and by definition of conditional probability, it'll be \(P('spam' 鈭� 'free') / P('free') = 0.0357 / 0.0475 = 0.75 = 75\% \).

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Most popular questions from this chapter

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