Question

An email system sends incoming mail to either the In-Folder (I) or the Trash Folder (T). You classify incoming mail as Useful (U), in which case you want it sent to I, or as a Nuisance (N) in which case you would like it sent to T. If incoming mail is U, the system sends it to T with probability 0.1. If the incoming mail is N, the system sends it to I with probability 0.05. Suppose a proportion 0.35 of your incoming mail is N.

What is the probability that an incoming mail is sent to T?

What is the probability that an incoming mail is U given that it is sent to T?

Answer 1

P[ the system sends it to T with probability | If incoming mail is U ] = 0.1

P[ the system sends it to I | If the incoming mail is N ] = 0.05.

P[ the system sends it to T | If the incoming mail is N ] = 1 - P[ the system sends it to I | If the incoming mail is N ] = 1 - 0.05 = 0.95

P[ incoming mail is N ] = 0.35

P[ incoming mail is U ] = 1 - incoming mail is N = 1 - 0.35 = 0.65

What is the probability that an incoming mail is sent to T = P[ T ]

P[ T ] = P[ T ^ U ] + P[ T ^ N ]

P[ T ^ U ] = P[ the system sends it to T with probability | If incoming mail is U ]*P[ incoming mail is U ]

P[ T ^ U ] = 0.1*0.65 = 0.065

P[ T ^ N ] = P[ the system sends it to T with probability | If incoming mail is N ]*P[ incoming mail is N ]

P[ T ^ N ] = 0.95*0.35 = 0.3325

P[ T ] = 0.065 + 0.3325

P[ T ] = 0.3975

Probability that an incoming mail is sent to T = 0.3975

What is the probability that an incoming mail is U given that it is sent to T = P[ U | T ]

P[ U | T ] = P[ T ^ U ] / P[ T ]

P[ U | T ] = 0.065/0.3975

P[ U | T ] = 0.1635