Question

Xiku Road has n1,n2,n3, and n4 houses with 1,2,3, and 4 occupants, respectively. Two random selection without replacement strategies are being contemplated to obtain a sample of the residents. In the first strategy, residents are selected with equal probability. In the second strategy, houses are first randomly selected and then residents from these houses are selected.

Work out, in terms of n1,n2,n3, and n4, the conditional probability of a resident from a 3-occupant house being selected given that the first selection came from a 4-occupant residence under both strategies.

Answer 1

First strategy-

Residents are selected with equal probability.

During first selection there are residents and we can select a resident from a 4-occupant house in ways.

During second selection there are residents and we can select a resident from a 3-occupant house in ways.

So, required conditional probability is given by

Second strategy-

Houses are selected first and then residents were selected.

During first selection there are houses and we can select a house from a 4-occupant houses in ways. Then we select a resident out of 4 residents of that selected house.

During second selection there are still houses as other members (remaining 3) of previously selected house may be selected and we can select a house from a 3-occupant house in ways. Then we select a resident out of 3 residents of that selected house.

So, required probability is given by