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Sampling With or Without Replacement

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Sampling can be done in a couple of different ways. One question that arises when sampling is, "After we select an object and record the measurement of attribute we're studying, what do we do with the object?" There are two options: we can replace the object into the pool of objects that we are sampling from, or we can choose to not replace the object.

Effect on Probabilities

To see how replacement affects the calculation of probabilities, consider the following example. What is the probability of drawing two aces from a standard deck of cards?

There are four aces and 52 cards total, so the probability of drawing one ace is 4/52. If we replace this card and draw again, then the probability is again 4/52. These events are independent, so we multiply the probabilities (4/52) x (4/52) = 1/169, or approximately 0.592%.

Now we will compare this to the situation when we do not replace the cards. The probability of drawing an ace on the first draw is still 4/52. For the second card, we assume that an ace has been already drawn. There are now three remaining out of a total of 51 cards. The probability of drawing two aces without replacement is (4/52) x (3/51) = 1/221, or about 0.425%.

Other Applications

There are other instances where we need to consider whether to sample with or without replacement. On example of this is bootstrapping. This statistical technique falls under the heading of a resampling technique.

In bootstrapping we start with a statistical sample of a population. We then use computer software to compute bootstrap samples. In other words, the computer restamples with replacement from the initial sample.

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