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 collected. Is the fact that the second selected fly black independent of the first one being black? Show your calculations. Suppose now 8, 000 are grey. A sample of 2 flies is selected. Is the fact that the second selected fly black inpependent of the first one being black? Show your calculations.

Solution

a)

If they are independent, then

P(second is black|first is black) = P(second is black|first is not black)

Here,

P(second is black|first is black) = 4/24 = 1/6 or 0.166667

P(second is black|first is not black) = 5/24 or 0.208333333

As they are not equal, then NO, THEY ARE NOT INDEPENDENT.

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b)

There are now 8000 grey, so proportionally, there are now 2000 black.

If they are independent, then

P(second is black|first is black) = P(second is black|first is not black)

Here,

P(second is black|first is black) = 1999/9999 = 0.199919992

P(second is black|first is not black) = 2000/9999 or 0.200020002

As they very close to being equal, then THEY CAN BE APPROXIMATED AS INDEPENDENT.