Scaling Random variables by a constant First, let me state the problem:
"Customers at Fred's cafe win a 100 dollar prize if their cash receipts show a star on each of the five consecutive days Monday...Friday in any one week. The cash register is programmed to print stars on a randomly select 10% of the receipts. If Mark eats at Fred's once each day for four consecutive weeks and the appearance of stars is an independent process, what is the standard deviation of $X$, where $X$ is the number of dollars won by Mark in the four-week period"
The probability model that can best solve this would be a Binomial with parameters $n = 4$ and $p = 0.1^5$. 
Folks can quickly solve this creating a random variable $Y = 100X$ and just getting the Variance of the binomial and taking the root. 
The variance for Binomial is $np(1-p)$ 
$$
\operatorname{Var}[100X] = 100^2 \operatorname{Var}[X].
$$
This method seems significantly more peaceful than finding the second moment and first moment of this new random variable $Y$. 
My issue comes from this allowance of multiplying $X$ by $100$ for the expected payout for this particular problem. How is it that we translate this word problem in such a way? When $n = 0$ this $100$ shouldn't be getting multiplied by the probability I thought. 
Can someone help me make this jump?
Thanks a lot!

$X$ is a binomial, that is it's the sum of $0$ and $1$, $0$ if you don't win and $1$ if you win. But you want to find the variance of the payoff, and in that case you get $0$ if you don't win and $100$ if you win. This is exactly $100X$, so that what you're looking for
