Consider a probability distribution obtained by rolling a die a hundred times. The mean is calculated using its formula.

As n increases, the mean value fluctuates, but as seen in this graph of mean versus the number of trials, the mean gradually approaches a constant value with increasing trials.  

The expected value of a random variable is the mean value as the sample size grows to infinity. In simple words, it is the long-run average of the results.

So, its formula is similar to that of the mean.

The concept of expected value is useful in decision theory. If one bets ten dollars on number 8 in roulette, there are 37 of 38 chances to lose and one of 38 chances to win.

If the winning money on the table is 360 dollars, the net gain on this small chance event would be 350 dollars.

The product of the random variable, with its probability, is summed to obtain the expected value.

This number tells us that one can expect to lose 53 cents for every ten-dollar bet.