Question

5% of the viewers in Mumbai watch all three News programs, 8% of them watch India at 9 and India Decides at 9 and 6% watch both India Decides at 9 and News Hour. Probability that a respondent of Mumbai will watch only India Decides at 9 is

• A. 0.164
• B. 0.094
• C. 0.220
• D. 0.2
• A. 0.45
• B. 0.24
• C. 0.340
• D. 0.66
• A. 0.66
• B. 0.82
• C. 0.78
• D. Data is insufficient

Answer 1

The Correct Option is B

Step 1: Understanding the problem.
We are asked to find the probability that a respondent from Mumbai will watch only "India Decides at 9." We are given certain probabilities for those who watch multiple programs. We need to use this information to calculate the probability for the specific case of watching only "India Decides at 9."
Step 2: Given Information.
- 5% of the viewers in Mumbai watch all three News programs.
- 8% of the viewers watch India at 9 and India Decides at 9.
- 6% of the viewers watch both India Decides at 9 and News Hour.
Let the total number of viewers in Mumbai be 100%. We can break the probabilities into different categories using the principle of inclusion and exclusion.
Step 3: Define variables.
- Let \( P(A \cap B \cap C) = 0.05 \) be the probability that viewers watch all three programs.
- Let \( P(A \cap B) = 0.08 \) be the probability that viewers watch both India at 9 and India Decides at 9.
- Let \( P(B \cap C) = 0.06 \) be the probability that viewers watch both India Decides at 9 and News Hour.
We are tasked with finding the probability that a viewer watches only "India Decides at 9," which is represented as \( P(B \setminus (A \cup C)) \), the probability of watching India Decides at 9 but not the other two programs.
Step 4: Applying the inclusion-exclusion principle.
The probability of watching both India at 9 and India Decides at 9 but not News Hour is: \[ P(A \cap B \setminus C) = P(A \cap B) - P(A \cap B \cap C) = 0.08 - 0.05 = 0.03 \] Similarly, the probability of watching both India Decides at 9 and News Hour but not India at 9 is: \[ P(B \cap C \setminus A) = P(B \cap C) - P(A \cap B \cap C) = 0.06 - 0.05 = 0.01 \] Thus, the probability of watching only India Decides at 9 is: \[ P(B \setminus (A \cup C)) = P(B) - P(A \cap B) - P(B \cap C) + P(A \cap B \cap C) \] We know that the probability of watching India Decides at 9 is the sum of the viewers who watch only that program, those who watch both India at 9 and India Decides at 9, and those who watch both India Decides at 9 and News Hour: \[ P(B) = P(A \cap B \setminus C) + P(B \cap C \setminus A) + P(A \cap B \cap C) = 0.03 + 0.01 + 0.05 = 0.09 \] Thus, the probability of watching only India Decides at 9 is: \[ P(B \setminus (A \cup C)) = 0.09 - 0.03 - 0.01 = 0.094 \] Step 5: Conclusion.
The correct answer is (B) 0.094, which is the probability that a respondent from Mumbai will watch only "India Decides at 9."

Answer 2

Step 1: Understanding the question.
The question asks for the probability that a respondent in Ahmedabad will not watch any of the three news programs. The total number of respondents in Ahmedabad is given as 1000. The number of viewers for each program is provided in the table.
Step 2: Calculating the number of viewers who watch at least one program.
We are given the number of viewers for each program:
- India at 9 (CNN IBN): 100
- India Decides at 9 (NDTV): 60
- News Hour (Times Now): 180
The probability of watching at least one of the programs is the sum of the viewers who watch the programs minus any overlaps. However, we are not given the overlapping viewers, so we will assume no overlaps and calculate the total as follows: \[ \text{Total viewers} = 100 + 60 + 180 = 340 \] Step 3: Probability of not watching any program.
The total number of viewers is 1000, and the number of viewers who watched at least one program is 340. Therefore, the number of viewers who did not watch any program is: \[ \text{Viewers not watching any program} = 1000 - 340 = 660 \] Thus, the probability that a respondent from Ahmedabad will not watch any of the three programs is: \[ P(\text{not watching any program}) = \frac{660}{1000} = 0.340 \] Step 4: Conclusion.
The correct answer is (C) 0.340.

Answer 3

Step 1: Understanding the question.
The question asks for the probability that a respondent in Mumbai or Pune will not watch any of the three news programs. However, the number of viewers who watched each program in these cities is given, but we do not have the data about the overlap between respondents in Mumbai and Pune, or the total number of viewers in these cities. Without this data, we cannot calculate the probability precisely.
Step 2: Conclusion.
Since we do not have the required data about overlaps or the total viewers in Mumbai and Pune, the correct answer is (D) Data is insufficient.