An e-mail filter is planned to separate valid e-mails from spam. The word f ree occurs in 60% of the spam messages and only 4% of the valid messages. Also, 20% of the messages are spam. Find the probability that the message is valid given that it does not contain the word f ree.

  (2, 4π/15), (2, 14π/15), (2, 24π/15)

Step-by-step explanation:

DeMoivre's theorem tells you the n-th root of a complex number in polar form is ...

  (magnitude, angle)^(1/n) = (magnitude^(1/n), (angle +2kπ)/n) for k = 0 to n-1.

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Your number has a magnitude of 8, so the cube root of that is 2.

Your number has an angle of (4π/5+2kπ), so one third of that is ...

  (π/3)(4/5 +2k) . . . for k = 0, 1, 2

Then the cube roots are (magnitude, angle) ...

  {(2, 4π/15), (2, 14π/15), (2, 24π/15)}

Of course, you can write (magnitude, angle) in CIS form as ...

  magnitude(cos(angle) +i·sin(angle))

as may be required by your grader.

Comment on complex number notation

The notation used in my engineering courses was fairly practical. A complex number could be written as a+bi or as magnitude∠angle. We didn't waste effort writing it as magnitude(cos(angle) +i·sin(angle)) and we avoided the confusion associated with different interpretations of an ordered pair.