An e-mail lter is planned to separate valid e-mails from spam. The word free occurs in 50% of the spam messages and only 3% of the valid messages. Also, 20% of the messages are spam. Determine the following probabilities: (a) The message contains the word free. (b) The message is spam given that it contains free. (c) The message is valid given that it does not contain free.

Apinhole camera is made out of a light-tight box with a piece of film attached to one side and pinhole on the opposite side. the optimum diameter d(in millimeters) of the pinhole can be modeled by d=1.9[ (5.5 x 10^-4) l) ] ^1/2, where l is the length of the camera box (in millimeters). what is the optimum diameter for a pinhole camera if the camera box has a length of 10 centimeters

Kayla made observations about the sellin price of a new brand of coffee that sold in the three different sized bags she recorded those observations in the following table 6 is $2.10 8 is $2.80 and 16 is to $5.60 use the relationship to predict the cost of a 20oz bag of coffee.

The height of a plant, in inches, p years after planting it is given by the polynomial function r(p) = -2p% + 270p. find the height of the plant when p = 90 years.