Answer:
e) P ( R' | T ) = 0.62857
f) P ( T' | N ) = 0.32258
g) P ( R' | N ) = 0.70968
h) P ( T' | N&R ) = 5/6
Step-by-step explanation:
Given:
- The probability of Television as news source P ( T ) = 0.7
- The probability of Newspaper as news source P ( N ) = 0.62
- The probability of Radio as news source P ( R ) = 0.46
- The probability of Television & Newspaper as news source P (T&N) = 0.42
- The probability of Television & Radio as news source P (T&R) = 0.26
- The probability of Radio & Newspaper as news source P (R&N) = 0.18
- The probability of all 3 as news source P ( T & R & N ) = 0.03
Find:
Given that TV is a news source, what is the probability that radio is not a news source?
Given that newspaper is a news source, what is the probability that TV is not a news source?
Given that newspaper is a news source, what is the probability that radio is not a news source?
Given that both newspaper and radio are news sources, what is the probability that TV is not a news source?
Solution:
- We will first compute the individual probability of each event occurring alone.
P (Television is the only news source) = P(T) - P(T&R) - P(T&N) + P(T&N&R)
P ( Only Television ) = P ( only T ) = 0.7 - 0.42 - 0.26 + 0.03 = 0.05
P (Newspaper is the only news source) = P(N)-P(N&R)-P(T&N)+P(T&N&R)
P ( Only Newspaper ) = P ( only N ) = 0.62 - 0.18 - 0.42 + 0.03 = 0.05
P (Radio is the only news source) = P(R) - P(T&R) - P(N&R) + P(T&N&R)
P ( Only Radio ) = P ( only R ) = 0.46 - 0.26 - 0.18 + 0.03 = 0.05
- Now for part e)
We are asked for a conditional probability of the form as follows:
P ( R' | T ) = P ( R' & T ) / P ( T )
First compute the probability that next news source is not Radio provided it is already a source of TV.
P ( R' & T ) = P( only T ) + P ( only T & N ) = 0.05 + 0.42 - 0.03 = 0.44
Hence,
P ( R' | T ) = 0.44 / 0.7
P ( R' | T ) = 0.62857
- Now for part f)
We are asked for a conditional probability of the form as follows:
P ( T' | N ) = P ( T' & N ) / P ( N )
First compute the probability that next news source is not TV provided it is already a source of Newspaper.
P ( T' & N ) = P( only N ) + P ( only R & N ) = 0.05 + 0.18 - 0.03 = 0.2
Hence,
P ( T' | N ) = 0.2 / 0.62
P ( T' | N ) = 0.32258
- Now for part g)
We are asked for a conditional probability of the form as follows:
P ( R' | N ) = P ( R' & N ) / P ( N )
First compute the probability that next news source is not Radio provided it is already a source of Newspaper.
P ( R' & N ) = P( only N ) + P ( only T & N ) = 0.05 + 0.42 - 0.03 = 0.44
Hence,
P ( R' | N ) = 0.44 / 0.62
P ( R' | N ) = 0.70968
- Now for part h)
We are asked for a conditional probability of the form as follows:
P ( T' | N&R ) = P ( T' & N & R) / P ( N & R )
First compute the probability that next news source is not TV provided it is already a source of both radio and Newspaper.
P ( T' & N & R) = P ( only N & R ) = 0.18 - 0.03 = 0.15
Hence,
P ( T' | N&R ) = 0.15 / 0.18
P ( T' | N&R ) = 5/6