Theoretical vs Empirical Probability:

Theoretical probability from a coin toss is 0.5 heads and 0.5 tails. Now, consider a brief experiment where you toss a coin four times and record all the four outcomes. Lets say, heads is one of the four outcomes and tails is the outcome, the remaining three times. So, from the experiment, the empirical probability is 0.25 for heads and 0.75 for tails. While the empirical probability is a valid outcome within the narrow frame of the experiment, any generalizations based on the experiment will be flawed and invalid. Similarly, theoretical probability has proven to be invalid within the frame of this particular experiment. In other words, empirical probability has greater internal validity with in the context of the experiment, however theoretical probability has greater external validity in terms of generalization.

When Theoretical Meets Empirical:

Lets extend our coin toss experiment further to 100 or even 1000 coin tosses. The theoretical probability will remain at 0.5 for heads, and 0.5 for tails. However, it is highly likely that the experimental probability will move further away from the results of the earlier experiment where the probability of heads was 0.25 and the probability of tails was 0.75. In fact, from a 100 or 1000 coin tosses, it is highly likely the probability of heads and tails will be closer to 0.5 respectively. This is because with repeated experimental measures, the empirical probability will be closer to theoretical probability.

Subjective Probability:

Mathematically or statistically least valid type of probability is the subjective probability. It does not necessarily involve logic or coherence. However that does not mean that it has no utility or validity in the real world. For instance, financial experts and analysts use subjective probability to predict how the markets are going to behave the next day or to recommend that stocks that are worth investing in. Subjective probability has utility when the outcome of the event is dependent on factors that are not quantitatively measurable. For instance, sports broadcasters often predict outcomes of a game based on subjective factors such as current player form, playing conditions or psychological edge, variables that may not necessarily always be factored in estimating empirical or theoretical probability of an outcome.