When we toss a coin, there is no way to predict which face (head or tail) will be facing up. There are simply too many factors that determined the final result. A tiny particle moves irregularly on the surface of some water, again, there are too many factors determine the movement of the particle. Since we can not precisely predict some events like these, we have to assume the randomness of the result. Probability is the science of randomness.

Statistics to probability is like egg and chicken, in the sense that no egg without chicken. But also it is hard to tell, which concepts comes out first. We say, if we flip a coin, the probability that the head up is 0.5. But this claim seems unfounded without doing some statistical experiments.

Here is the first question for you, if you are a beginner:

Flip a fair coin, if you get head, you win 2 dollars, if you get a tail, you lose 1 dollar. If you flip the coin 10 times, how much would you expect to get?

You might say, I do it 10 times, so it is, on average, most likely that 5 times, I got head, another 5 times, I got tail. Also intuitively, this is equivalent to flip 10 coins and see the result immediately. So I would expect to get

$5\times 2 -5\times 1 = 5$ dollars.

Then, you are right, basically! If let $X$ denote the (random) result of the dollars you earned for one flip, you have just calculated $E(10X)$, which is also equal $10E(X)$. Or using a fancy word, you have calculated the expectation of $10X$.

While in some statistics books it take over a hundred pages to get you here, you can learn from us in a few minutes. Of course, we can also help you on graduate level probability theory on martingales, Brownian motion, stochastic differential equations.