Probability
Theory

Probability Theory is the mathematical study of randomness. This theory deals with the possible outcomes of an event. It must be possible to list every outcome that can occur, and we must be able to state the expected relative frequencies of these outcomes. It is the method of assigning relative frequencies to each of the possible outcomes.
Probability
Described
in
Long-term
Relative
Frequency
As it was mentioned before, tossing a coin thousands of times would give a probability of getting heads half of the time. In other words, the observed proportion of heads becomes close to ½ (It is said that the probability of a head on any single toss is ½). In intuitive terms it can be concluded that probability is long-term relative frequency. Having that in mind, correct probability can only be determined by actually observing many tosses of the coin to see if about ½ of the outcomes are heads. We, however, can not observe a probability exactly, simply because we could always continue tossing the coin. Therefore, it can be concluded that mathematical probability is an idealization based on imagining what will happen to the relative frequencies in an indefinite amount of trials.

Example 1:
To demonstrate the fact that relative frequency stabilizes in the long run, here is an example showing the relative frequency of getting exactly two tails when tossing four coins at the same time:

 Trial Outcome Relative frequency of exactly 2 tails 1 3 tails 0/1 = 0 2 0 tails 0/2 = 0 3 0 tails 0/3 = 0 4 2 tails 1/4 = 0.25 5 4 tails 1/5 = 0.20 6 2 tails 2/6= 0.33

It is clearly seen that the relative frequency is 0 for the first three trials. After the fourth trial resulting in two tails, the relative frequency changes to ¼ or 0.25 (because 1 out of 4 trials have resulted in two tails). The relative frequency usually becomes more exact after the first 100 trials and it stabilizes after more repetitions are made.

Personal
Probability
Long-term frequency is not the only way of interpreting probability. We can have a personal opinion about the next outcome of an event such as a coin toss. I can say that my personal probability of a head in the next toss is ½. Your personal probability may be different from mine. Personal probability sets us free from figuring out the outcome from many repetitions. Therefore, personal probability allows us to assign a probability to one-time events such as a golf tournament. Although we can have a personal probability about an event, we should also keep in mind the long-term relative frequency of chance.