Question

A survey is conducted to find out whether people in metropolitan areas obtain their news from television (Event T), a newspaper (Event N), or radio (Event R). The results show that 60% of people get news on television, 60% from newspapers, 50% from radio, 30% from television and newspapers, 25% from television and radio, 30% from newspapers and ratio, and 15% from television, newspapers and radio. a) What is the proportion of those who obtain news from television, but not newspapers? 2 b) What is the proportion of those who do not obtain news from either television or radio? c) What is the proportion of those who do not obtain news from any of these three sources? d) Given that radio is a news source, what is the probability that a newspaper is also a news source?

Answer 1

Let T be the event that denotes news from television.

Let N be the event that denotes news from a newspaper.

Let R be the event that denotes news from radio.

Given, P(T) = 0.60 ; P(N) = 0.60 ; P(R) = 0.50 ; P(T N) = 0.30 ; P(T R) = 0.25 ; P(N R) = 0.30 ; P(T N R) = 0.15

a) Answer :

P(T) - P(T N) = 0.60 - 0.30

                         = 0.30

Therefore, the proportion of those who obtain news from television, but not newspaper is 30%.

b) Answer :

P(T R)c = 1 - P(T R)    ............('c' denotes complement)

                 = 1 - (P(T) + P(R) - P(T R))     .................(using the addition rule of probability)

                 = 1 - (0.60 + 0.50 - 0.25)

                 = 0.15

Therefore, the proportion of those who do not obtain news from either television or radio is 15%.

c) Answer :

P(T N R)c = 1 - P(T N R)      ............('c' denotes complement)

                        = 1 - (P(T) + P(N) + P(R) - P(T N) - P(T R) - P(N R) + P(T N R))   ........(using the addition rule of probability)

                        = 1 - (0.60 + 0.60 + 0.50 - 0.30 - 0.25 - 0.30 + 0.15)

                        = 0

Therefore, the proportion of those who do not obtain news from any of these three sources is 0%.

d) Answer :

P(N | R) = P(N R) / P(R)    ...........(conditional probability)

              = 0.30 / 0.50

              = 0.60

Therefore, the probability that a newspaper is also a news source given that radio is a news source is 0.60