# What does the words "with replacement " & "without replacement " mean in an probability question??

Arun
25750 Points
6 years ago

.  Suppose that we are randomly choosing two people from a city with population of 50,000, of which 30,000 of these people are female.

If we sample with replacement, then the probability of choosing a female on the first selection is given by 30000/50000 = 60%.  The probability of a female on the second selection is still 60%.  The probability of both people being female is 0.6 x 0.6 = 0.36.

If we sample without replacement then the first probability is unaffected.  The second probability is now 29999/49999 = 0.5999919998. . ., which is extremely close to 60%.  The probability that both are female is 0.6 x 0.5999919998 = 0.359995.

The probabilities are technically different, however they are close enough to be nearly indistinguishable.  For this reason, many times even though we sample without replacement, we treat the selection of each individual as if they are independent of the other individuals in the sample.