In a survey of a group of people the following results are obtained. 40% own a Tape recorder, but not a colour T.V. 10% own a colour T.V. but not a Tape recorder. 20% own neither colour T.V. nor a Tape recorder. 30% own both colour T.V. and a Tape recorder.

1. In a survey of a group of people the following results are obtained.
40% own a Tape recorder, but not a colour T.V.
10% own a colour T.V. but not a Tape recorder.
20% own neither colour T.V. nor a Tape recorder.
30% own both colour T.V. and a Tape recorder.
Required
(i) What is the percentage of people who own either a colour T.V.
or a Tape recorder or both.
(ii) What is the percentage of people who own a Tape recorder
among the colour T.V. owners.
Solution
𝑻𝑹 = 𝑶𝒘𝒏 𝒂 𝒕𝒂𝒑𝒆 𝒓𝒆𝒄𝒐𝒓𝒅𝒆𝒓
𝑻
̅𝑹 = 𝑵𝒐𝒕 𝒐𝒘𝒏 𝒂 𝒕𝒂𝒑𝒆 𝒓𝒆𝒄𝒐𝒓𝒅𝒆𝒓
𝑻𝑽 = 𝑶𝒘𝒏 𝒂 𝒄𝒐𝒍𝒐𝒖𝒓 𝑻. 𝑽.
𝑻
̅𝑽 = 𝑵𝒐𝒕 𝒐𝒘𝒏 𝒂 𝒄𝒐𝒍𝒐𝒖𝒓 𝑻. 𝑽.
𝑷(𝑻𝑹 ∩ 𝑻
̅𝑽) = 𝟎. 𝟒𝟎
𝑷(𝑻𝑽 ∩ 𝑻
̅𝑹) = 𝟎. 𝟏𝟎
𝑷(𝑻
̅𝑽 ∩ 𝑻
̅𝑹) = 𝟎. 𝟐𝟎
𝑷(𝑻𝑽 ∩ 𝑻𝑹) = 𝟎. 𝟑𝟎
First to find 𝑷(𝑻𝑹) and 𝑷(𝑻𝑽)

2. As we know
𝑷(𝑻𝑹 ∩ 𝑻
̅𝑽) = 𝑷(𝑻𝑹) − 𝑷(𝑻𝑽 ∩ 𝑻𝑹)
𝟎. 𝟒𝟎 = 𝑷(𝑻𝑹) − 𝟎.𝟑𝟎
𝑷(𝑻𝑹) = 𝟎. 𝟒𝟎 + 𝟎.𝟑𝟎 = 𝟎.𝟕𝟎
And
𝑷(𝑻𝑽 ∩ 𝑻
̅𝑹) = 𝑷(𝑻𝑽) − 𝑷(𝑻𝑽 ∩ 𝑻𝑹)
𝟎. 𝟏𝟎 = 𝑷(𝑻𝑽) − 𝟎.𝟑𝟎
𝑷(𝑻𝑽) = 𝟎. 𝟏𝟎 + 𝟎.𝟑𝟎 = 𝟎.𝟒𝟎
(i) 𝑷(𝑻𝑽 ∪ 𝑻𝑹) = 𝑷(𝑻𝑽) + 𝑷(𝑻𝑹) − 𝑷(𝑻𝑽 ∩ 𝑻𝑹)
𝑷(𝑻𝑽 ∪ 𝑻𝑹) = 𝟎. 𝟒𝟎 + 𝟎.𝟕𝟎 − 𝟎. 𝟑𝟎 = 𝟎. 𝟖𝟎 = 𝟖𝟎%
(ii) 𝑷(𝑻𝑽 ∩ 𝑻𝑹) = 𝑷(𝑻𝑽) × 𝑷(𝑻𝑹)
𝑷(𝑻𝑹) =
𝑷(𝑻𝑽 ∩ 𝑻𝑹)
𝑷(𝑻𝑽)
𝑷(𝑻𝑹) =
𝟎. 𝟑𝟎
𝟎. 𝟒𝟎
= 𝟎. 𝟕𝟓 = 𝟕𝟓%