Sampling with or without replacement

Let's now assume that there is a total of n items in the bucket and we must pick r of them. Then, let R = {1, 2,…, r} be the list of items picked and let N = {1, 2, …, n} be the total number of items. This can be written as a function, as follows:

Here, f(i) is the i^th item.

Sampling with replacement is when we pick an item at random and then put it back so that the item can be picked again. 

However, sampling without replacement refers to when we choose an item and don't put it back, so we cannot pick it again. Let's see an example of both.

Say we need to open the door to our office and we have a bag containing n keys; they all look identical, so there's no way of differentiating between them. 

The first time we try picking a key, we replace each one after trying it, and we manage to find the correct key on the r^th trial, implying we got it wrong r-1 times. The probability is then as follows:

Now, we know that our earlier strategy wasn't the smartest, so this time we try it again but without replacement and eliminate each key that doesn't work. Now, the probability is as follows: