In an isolated housing area consisting of 50 households, the residents are allow

Question

In an isolated housing area consisting of 50 households, the residents are allowed to have at most one dog and at most one cat per household. Currently 25 households have a dog, 7 have both a cat and a dog, and 18 have neither a cat nor a dog. An experiment consists of randomly selecting a household and observing whether that household has a cat or a dog, or both. Using D to denote the event that the selected household has a dog and C to denote the event that the selected household has a cat, symbolically denote the following events and identify the number of households represented by each. a) The selected household has a dog but no cat. b) The selected household has a cat. c) The selected household has a cat but no dog. d) The selected household has a cat or a dog but not both.

Explanation / Answer

Let C denotes the Cat and D denotes the dog

Number of households having dog alone = Number of houses having dogs - number of houses having both dogs and cats

=> 25 - 7 = 18

Number of houses that have cats and dogs = 50 - number of houses having no pets = 50 - 18 = 32

P(CUD) = 32

P(D) = 25

P(C int D) = 7

P(CUD) = P(C) + P(D) - P(C int D)

32 = 25 + P(C) - 7

P(C) = 14 , hence there are 14 cats in the society

a) P(only dog) = 18

b) P(cat) = 14

c) P(cat but no dog) = 14 - 7 = 7

d) P(cat or dog but not both) = P(C) + P(D) - 2P(C int D) = 25 + 14 - 14 = 25

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