I am currently working through Stowe's Thermodynamics and Statistical Mechanics third chapter, titled "Systems with many elements". The first exercise problem is as follows:
Consider many systems, each having 100 rolled dice. Suppose that we are
interested in the number of dice per system showing sixes. For these systems,
calculate (a) the mean number of sixes and (b) the standard deviation about this value.
My first attempt at solving this was to note that the probability of getting $k$ sixes is $P(k \text{ sixes}) = {100 \choose k} \left( \frac{1}{6}\right)^k \left( \frac{5}{6} \right)^{100-k}$. Hence the the average number of sixes rolled is $\bar{k} = \sum_{n=0}^{100}n {100 \choose n} \left( \frac{1}{6}\right)^n \left( \frac{5}{6} \right)^{100-n}$. I opened excel and computed the above equation and obtained $\bar{k} \approx 16.7$.
I checked the solutions at the back of the textbook, and indeed that is the correct answer. However, my worry is that the second exercise requires us to do the same, but this time with systems composed of $10^8$ dice.
Obviously, there is no way I can use excel to do such a calculation, so I am not sure on how I could approximate a solution. I thought about using a Gaussian distribution, but I do not see how I could set it up so it tells me the specific number of sixes rolled. Any ideas?
Best Answer
Let $X_i$ denote the random variable which takes value 1 if the $i^{th}$ die showed a 6 (with probability 1/6) and takes value 0 if it didn't (with probability 5/6).
The number of dice which showed 6 is $Y = \sum_{i=1}^{100} X_i$, by definition.
The expected number of dice which showed 6 is $\mathbb{E}[Y] = \sum_{i=1}^{100} \mathbb{E}[X_i]$
From the definition of $X_i$, $\mathbb{E}[X_i] = {1 \over 6} \times 1 + {5 \over 6} \times 0 = {1 \over 6}$
Putting it all together, $\mathbb{E}[Y] = 100/6$
Bottomline, you don't need to compute the distribution to compute the expected value