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In a survey, 54 people were asked which political party they had voted for in the last three elections - Junior Cycle Mathematics - Question a - 2013
Question a
In a survey, 54 people were asked which political party they had voted for in the last three elections. The results are as follows: 30 had voted for the Conservativ... show full transcript
Worked Solution & Example Answer:In a survey, 54 people were asked which political party they had voted for in the last three elections - Junior Cycle Mathematics - Question a - 2013
Step 1
Represent the information in a Venn diagram.
Answer
To create the Venn diagram, we need to identify and place the intersections based on the provided data:
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Identify Individual Votes: Start with the number of people who voted for all three parties.
- The number of people who voted for all three parties (Conservatives, Liberals, Republicans) is 5.
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Fill in Pairwise Intersections: Next, we fill in the number of people who voted for two parties, taking into account those who also voted for all three:
- Conservatives and Liberals: 12 (total) - 5 (all three) = 7
- Liberals and Republicans: 9 (total) - 5 (all three) = 4
- Conservatives and Republicans: 8 (total) - 5 (all three) = 3
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Fill in the Individual Sets: Now, we can find out how many voted only for one party by subtracting these intersections from the total counts:
- Conservatives only: 30 (total) - (7 + 3 + 5) = 15
- Liberals only: 22 (total) - (7 + 4 + 5) = 6
- Republicans only: 22 (total) - (4 + 3 + 5) = 10
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Completing the Venn Diagram: We now place these values in the Venn diagram as follows:
- Let C be the set of Conservatives, L be the set of Liberals, and R be the set of Republicans. The diagram will have:
- C: 15 only, 7 (CL), 3 (CR), 5 (CLR)
- L: 6 only, 7 (CL), 4 (LR), 5 (CLR)
- R: 10 only, 4 (LR), 3 (CR), 5 (CLR)
- Let C be the set of Conservatives, L be the set of Liberals, and R be the set of Republicans. The diagram will have:
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Universal Set (U): The total number surveyed is 54, which needs to be displayed outside the sets.
Step 2
If one person is chosen at random, what is the probability that the person chosen did not vote in any of the three elections?
Answer
To determine the probability, first, we need to calculate how many people did not vote in any of the three elections:
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Count those who voted: Total surveyed (54) minus those who voted (calculated from the Venn diagram):
- People who voted = 15 + 6 + 10 + 7 + 4 + 3 + 5 = 50.
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Calculate non-voters: Thus, people who did not vote = 54 - 50 = 4.
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Calculate Probability: The probability that a randomly chosen person did not vote is given by:
P(not voting) = ( \frac{4}{54} = \frac{2}{27} )
Step 3
If one person is chosen at random, what is the probability that the person chosen voted for at least two different parties?
Answer
To find the probability of choosing someone who voted for at least two parties, we count those who voted for two and three parties:
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Count the Voters for Two or More Parties:
- Those who voted for exactly two parties = 7 (CL) + 4 (LR) + 3 (CR) = 14.
- Those who voted for all three parties = 5.
- Total = 14 + 5 = 19.
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Calculate Probability: Therefore, the probability is:
P(voted for at least 2 parties) = ( \frac{19}{54} )
Step 4
If one person is chosen at random, what is the probability that the person chosen voted for the same party in all three elections?
Answer
To find this probability, we need to know how many voted for the same party in all three elections:
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Voters for the same party: This includes only those who voted for one party across all elections. Based on the overlapping counts:
- Conservatives: 15 for only Conservatives + 7 + 3 + 5 = 30.
- Liberals: 6 for only Liberals + 7 + 4 + 5 = 22.
- Republicans: 10 for only Republicans + 4 + 3 + 5 = 22.
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Voters for Each Party in All Elections: Since we have the overlaps, we need to see only those who voted for the same party:
- All for one party: 15 (Conservatives) + 6 (Liberals) + 10 (Republicans). Total = 15 + 6 + 10 = 31.
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Calculate Probability: Thus, the probability that a randomly chosen person voted for the same party in all three elections is:
P(same party in all elections) = ( \frac{31}{54} )