Difference in the variance between multiply a random variable and the sum of the same random variable I suppose this is a pretty simple question but i can't figure out what i don't understand
Let's say we have X, a random variable representing the result of flipping a coin, 1 for head and -1 for tail. 
If we have Y the sum of 10 throws and if we assume each throw is independent, we have $Var(Y) = 10 * Var(X) = 10$
But I also saw that $ Var(10 * X) = 10^2 * Var(X) = 100$
Why can't we consider that $Var(Y) = Var(10 * X)$ ? 
I know it works mathematically but i would like to understand why it's not the same. 
 A: The random variable $Y$ is the sum of 10 coin flips.
$$Y = X_1 + \dots + X_{10}$$
The coin flips are independent, therefore, we can write the variance of $Y$ as
$$Var(Y) = Var(X_1 + \dots + X_{10}) = Var(X_1) + \dots + Var(X_{10}) = 10*Var(X_1)$$
The last equality comes because each flip has the same probability mass function and they have the same variance.
Each flip corresponds to an independent random variable with its own probability mass function. Each flip is a different random event and you cannot simply treat them the same.
For example, to be $X_1 = X_2$, the outcome of the first flip should be the same as the second flip's outcome. That is to say, two coin flips have to be correlated perfectly. But it violates the assumption of flip's independence. Because each flip is independent of each other, it's possible that the first coin flip results in the head but the second coin flip's outcome is the tail. And in the case, $ X_{1} \neq X_{2} $ because $X_1 = 1$ and $X_2 = -1$.
That is to say, 
$$ X_{1} \neq X_{2} \neq \dots \neq X_{10}$$ 
and,
$$Y = X_1 + \dots + X_{10} \neq 10*X_1$$
