Empirical probability of an event is estimated by observing results of actual experiments. For example, if you wanted to know the probability of the number 6 turning up when a fair die is rolled you would have to roll the die a large number of times and see how many times the number 6 turns up.

The empirical probability of an event is given by number of times the event occurs divided by the total number of incidents observed.

Theoretical probability on the other hand is given by the number of ways the particular event can occur divided by the total number of possible outcomes. For example the theoretical probability of the number 6 turning up when a fair die is rolled is given by 1/6 as the total number of outcomes is 6 and 1 of them is 6.

The empirical probability of an event comes closer to the theoretical probability as the number of observations becomes larger. The probability could be skewed if a small number of observations are available; in reality only if the number of observations is infinite does the empirical probability become equal to the theoretical probability.

Theoretical probabilities are your calculation of what "should" happen, when you are able to describe all the equally likely outcomes. Then the theoretical probability is the number of "successes" (outcomes that fit the probability you're trying to find) divided by the number of equally likely outcomes. For example, if you roll a fair six-sided die, there are six equally likely outcomes: 1, 2, 3, 4, 5, and 6. The probability that you roll an odd number is 3/6 (equals 1/2) because three of those outcomes - 1, 3, and 5 - are odd, which makes them "successes."

Empirical probabilities are your calculation of what "did" happen in an experiment. Again, you put the number of successes in the numerator, and the denominator is the number of trials you did. If I roll a fair six-sided die ten times, one possible result is that I roll 4, 2, 5, 6, 3, 2, 3, 5, 2, 2. My empirical probability of rolling an odd number in this case is 4/10 (equals 2/5).

Notice that the empirical probability is not necessarily equal to the theoretical probability. Notice also that if you do the experiment again, you may well get a different (but still correct) empirical probability. The theoretical probability, on the other hand, has only one correct answer.

As for professions, consider industries like insurance or casinos and name some of the people who work for those companies. Probability theory is also the foundation of statistical inference, used by statisticians and researchers of all kinds.