Probability: Application Of "Expected Value"

Question

$\newcommand{\P}{\mathbb{P}}$$\newcommand{\E}{\mathbb{E}}$So, I was learning expected value today and I'm trying to understand the significance of calculating this term "Expected value".

In this simple example,

What is the expected value when we roll a fair die?

There are six possible outcomes: $1, 2, 3, 4, 5, 6$. Each of these has a probability of $1/6$ of occurring. Let $X$ represent the outcome of the experiment.

Therefore
$\P(X = 1) = 1/6$ (this means that the probability that the outcome of the experiment is $1$ is $1/6$),
$\P(X = 2) = 1/6$ (the probability that you throw a $2$ is $1/6$),
$\P(X = 3) = 1/6$ (the probability that you throw a $3$ is $1/6$),
$\P(X = 4) = 1/6$ (the probability that you throw a $4$ is $1/6$),
$\P(X = 5) = 1/6$ (the probability that you throw a $5$ is $1/6$),
$\P(X = 6) = 1/6$ (the probability that you throw a $6$ is $1/6$). $$\E(X) = 1\!\cdot\!\P(X = 1) + 2\!\cdot\!\P(X = 2) + 3\!\cdot\!\P(X = 3) + 4\!\cdot\!\P(X=4) + 5\!\cdot\!\P(X=5) + 6\!\cdot\!\P(X=6).$$
Therefore $$\E(X) = \frac{1}{6} + \frac{2}{6} + \frac{3}{6} + \frac{4}{6} + \frac{5}{6} + \frac{6}{6} = \frac{7}{2}$$ So the expectation is $3.5$. If you think about it, $3.5$ is halfway between the possible values the die can take and so this is what you should have expected.

Please enlighten me on the significance of this term and the motivation to calculate "Expected Value".

Answer 1

I'm going to quote Wikipedia here,

In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents. For example, the expected value in rolling a six-sided die is 3.5 because, roughly speaking, the average of all the numbers that come up in an extremely large number of rolls is very nearly always quite close to three and a half. Less roughly, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity.

So we are speaking of the long-run average. I have simulated 1000 rolls, and have plotted the mean after each roll. Notice that when we take the average of 1000 rolls, we are near 3.5. So, although, there is no face value 3.5 on a die, the average of many rolls can still certainly be near or exactly 3.5.