A lab has a population of 25 fruit flies of which 5 are blac

A lab has a population of 25 fruit flies, of which 5 are black and 20 are grey. A sample of 2 flies is selected. Is the fact that the second selected fly is black independent of the first one being black? Suppose now that there are 10,000 flies in the lab, of which 2,000 are black and 8,000 are grey. A sample of 2 flies is selected. Is the fact that the second selected fly is black independent of the first one being black?

Solution

(a) The selction of black fly in second turn is not independent of the fly selected in first turn.

This is because, there are limited number of flies of each color and the further the flies are selected with replacement. Therefore, once a fly selected in first turn is found to be black then the chances of selcting a black fly is dependent on first fly as the total number of flies left there will be less than one now.

(b). The same reasoning can be followed here that selection without replacement reduce the total number of flies for the second selection and hence the second selection is depenedent of first selection.