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.

This is so hard and it’s for a test

Ateam has p points. p = 3w + d w = wins d = draws a) a football team has 5 wins and 3 draws. how many points does the team have?

White shapes and black shapes are used in a game. some of the shapes are circles. while the rest are squares. the ratio of the number of white shapes to the number of black shapes is 5: 11. the ratio of the number of white cicrles to the number of white squares is 3: 7. the ratio of the number of black circles to the number of black squres is 3: 8. work out what fraction of all the shapes are circles.