A researcher has found old census data about Measles (M), Chickenpox (C), and Whooping cough (W) among 12-year-old children. In a group of 100 children: 31 had none of these diseases 2 had all three diseases 2 had Measles and Chickenpox, but not Whooping cough 6 had Whooping cough and Chickenpox 11 had at least two diseases 18 had Measles 40 had Chickenpox. (a) Use this data to fill in the Venn diagram. (b) Find the probability that a child chosen at random from the group had Chickenpox. (c) Complete the table. The table below shows 3 statements. Each statement is written in English and in set notation. Statement 1 6 had Whooping cough and Chickenpox Statement 2 36 had Chickenpox but not Measles Statement 3 2 had Measles and Chickenpox but not Whooping cough.

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A researcher has found old census data about Measles (M), Chickenpox (C), and Whooping cough (W) among 12-year-old children - Junior Cycle Mathematics - Question 10 - 2015

Question 10

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A researcher has found old census data about Measles (M), Chickenpox (C), and Whooping cough (W) among 12-year-old children. In a group of 100 children: 31 had none... show full transcript

Worked Solution & Example Answer:A researcher has found old census data about Measles (M), Chickenpox (C), and Whooping cough (W) among 12-year-old children - Junior Cycle Mathematics - Question 10 - 2015

Step 1

Use this data to fill in the Venn diagram.

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Answer

To fill in the Venn diagram, we first identify the various subsets of children with diseases. Here's the breakdown:

Children with all three diseases (M, C, W): 2
Children with Measles and Chickenpox but not Whooping cough (M ∩ C ∩ W'): 2
Children with Whooping cough and Chickenpox (W ∩ C): 6 (which includes those with all three, so 6 - 2 = 4 will go in W ∩ C only)
Children with at least two diseases (calculated):
- Children with Measles and Chickenpox: 2 + 2 (those with all three) = 4
- Children with Whooping cough and Measles (W ∩ M): Not given directly, but can be deduced later.
Children with only Measles: 18 - 2 (M ∩ C) - 2 (M ∩ W) = 14
Children with Chickenpox only can be found as:
- 40 (total with Chickenpox) - 2 (M ∩ C) - 4 (W ∩ C) - 2 (those with all three) = 32
Children with none of the diseases: 31.

Thus filled, the counts in the Venn diagram would be:

Only whooping cough: 15
Only Chickenpox: 32
Only Measles: 14
Measles and Chickenpox only: 2
Whooping cough and Chickenpox only: 4
All three diseases: 2
None: 31.

Step 2

Find the probability that a child chosen at random from the group had Chickenpox.

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Answer

To find the probability that a child chosen at random had Chickenpox, we use the formula:
$P(C) = \frac{\text{Number of children with Chickenpox}}{\text{Total number of children}}$
In this case, there are 40 children with Chickenpox among 100 children:
$P(C) = \frac{40}{100} = \frac{2}{5}$
Thus, the probability is ( P(C) = \frac{2}{5} ).

Step 3

Complete the table.

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Answer

Statement 1:
6 had Whooping cough and Chickenpox.
Set notation:
$6 = \#(W \cap C)$
Statement 2:
36 had Chickenpox but not Measles.
Set notation:
$36 = \#(C \setminus M)$
Statement 3:
2 had Measles and Chickenpox but not Whooping cough.
Set notation:
$2 = \#(M \cap C \cap W')$

A researcher has found old census data about Measles (M), Chickenpox (C), and Whooping cough (W) among 12-year-old children - Junior Cycle Mathematics - Question 10 - 2015

Worked Solution & Example Answer:A researcher has found old census data about Measles (M), Chickenpox (C), and Whooping cough (W) among 12-year-old children - Junior Cycle Mathematics - Question 10 - 2015

Use this data to fill in the Venn diagram.

Find the probability that a child chosen at random from the group had Chickenpox.

Complete the table.

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