# Understanding Sampling with replacement

I am doing Stat110 and in the book, "Introduction to Probability" they give the following definition of sampling with replacement:

Theorem 1.4.5 (Sampling with replacement). Consider n objects and making k choices from them, one at a time with replacement (i.e., choosing a certain object does not preclude it from being chosen again). Then there are n^k possible outcomes.

I tried to use the above definition to find out the number of outcomes (sample size) in a roll of two fair 6-sided dice and 12 fair 6-sided dice. I know the answer is 36 in one case and 6^12 in the other but I cannot really understand how to use the above theorem to get this number.

For example, if I have two 6-sided dice then n = 2 and each die has 6 choices. So should it not be 2^6?

• Thanks Antoine. This is what I was looking for. Please write your response as the "answer" so I can set this question as "answered". Thanks again! – Rohit May 27 '15 at 9:53
• You're welcome. I have added the comment as an answer and deleted it as a comment as it is now redundant – Antoine Vernet May 27 '15 at 10:34

I think the issue here is that what the theorem call objects and choices are not what we intuitively call objects and choices.

The objects here are not the dice but each side of a die. So you have 6 objects in a die. The choices are then the number of time you choose among those objects.

Therefore, in you case, n is 6 and k is the number of time you roll the die or the number of dice you are rolling in one go.

Let us begin with the contradiction. Your finding of 2^6 cannot be correct because each of the six sides of the first die can be matched with any if the six sides from the second. As a result there are 6*6=36 sides, not 2^6=64.

But you did apply a formula. What did that formula mean? The formula of 2^6 implies you are making six draws from two objects. This formula would be great if you had two colored balls in a jar and drew from it six times, with replacement. The observed sequence of colors would be one of 2^6 possible sequences.

"With replacement" simply means that when you repeat the experiment (throw dice) the chances are the same as before.

Imagine that you had a magic dice: every time you throw it, the side disappears! Let's say you throw it once and get 4. Now 4 disappears, and the dice has only 5 sides. You throw it second time and get 1, now side 1 disappears and the dice has only 4 sides left. This magic dice is "without replacement". The normal dice throwing would be "with replacement" because every time you throw it's the same game: 6 sides, equal chances.

With two throws of a fair dice, the simplest way to visualize the trials is to imagine a square where each row is the first throw and the columns correspond to a second throw. There's 36 possible combinations of 2 throw outcomes of a 6-sided dice.

If you imagine a magic dice as described above, then after the first throw the row corresponding to the side disappears from the square, hence, it has fewer combinations possible, namely 6x5=30.