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59 families are asked whether they have a cat (C) or a dog (D) - OCR - GCSE Maths - Question 9 - 2020 - Paper 1
Question 9
59 families are asked whether they have a cat (C) or a dog (D). 26 only have a cat. 14 only have a dog. 11 have both a cat and a dog. (a) Show this information on ... show full transcript
Worked Solution & Example Answer:59 families are asked whether they have a cat (C) or a dog (D) - OCR - GCSE Maths - Question 9 - 2020 - Paper 1
Step 1
Show this information on the Venn diagram.
Answer
To represent the information on a Venn diagram:
- Families with only a cat (C): Place 26 in the section of the diagram representing families with only a cat.
- Families with only a dog (D): Place 14 in the section of the diagram representing families with only a dog.
- Families with both a cat and a dog: Place 11 in the overlapping section representing families that have both a cat and a dog.
Thus, the Venn diagram is populated as follows:
- C (only cat): 26
- D (only dog): 14
- CD (both cat and dog): 11
Step 2
How many of the families do not have a cat or a dog?
Answer
To find how many families do not have a cat or a dog, we first calculate the total number of families represented:
- Total families = 59
- Families with only a cat = 26
- Families with only a dog = 14
- Families with both cat and dog = 11
Now, we determine the families with at least one cat or dog:
Total with pets = 26 (only cat) + 14 (only dog) + 11 (both) = 51
Families without either = Total families - Families with pets
Families without either = 59 - 51 = 8
Thus, there are 8 families that do not have a cat or a dog.
Step 3
Step 4
Write down the probability that they have a dog.
Answer
To determine the probability that a randomly chosen family has a dog, we use the following formula:
Probability = rac{ ext{Number of families with a dog}}{ ext{Total number of families}}
From the Venn diagram:
- Families with only dogs = 14
- Families with both = 11
Total families with dogs = 14 + 11 = 25
Now calculating the probability:
Probability = rac{25}{59}
Thus, the probability that a randomly chosen family has a dog is rac{25}{59}.