Students should have some practical experience with tossing a set of M&M’s and recording the number of M&M’s that land up, showing the letter “M”. (See http://www.math.nmsu.edu/~breakingaway/Statistics/Lessons/M&Ms/M&Ms.html.)
Consider an experiment in which we toss 100 M&M’s many times, and for each outcome we record the number of “Ms”. What we observe and record are values of a random variable.
We call it a variable, and we may name it (for example, x) because we observe many different numbers. The range of x is the set of numbers between 0 and 100 (0 ≤ x ≤ 100), because these are all possible outcomes of our experiment.
It is a random variable because there is no way of predicting individual outcomes, and also the next outcome doesn’t depend on the previous ones. But at the same time, when we gather more and more data, we start observing regularity, which we call a probability distribution:
More than half of the time the number of “Ms” falls between 45 and 55. And the values outside the range {35, 65} are rare. The exact 50 (one half of a hundred) occur less often than one may expect, only 8% of the time.
This process was studied in detail and we can predict with excellent precision the frequency of each outcome when the experiment is repeated very many times. These predictions are called the theoretical probability distribution and are compared to the observed frequency distribution.
How many times would you need to toss 100 M&M’s to have a reasonable chance to see one hundred “Ms” in one trial?
Approximately: 1,000,000,000,000,000,000,000,000,000,000 times. It means that there is no chance that it will ever happen, but still it may happen.
Expected value
The expected value of the theoretical probability distribution in this tossing experiment is 50. Of course it is not an outcome you expect to see (50 “Ms” are observed approximately 8% of the time).
But when you toss M&Ms many times (for example, 500 or more) you may expect that the average of all outcomes will be close to 50.
So the expected value means: A number you may expect to be close to the average of all outcomes when the number of trials is really big.