7.2 Probability Theory

Probability is defined as the likelihood that the event will occur. Probability measures the uncertainty associated with the outcomes of a random experiment. Some other terms or words used in place of probability are chance, likelihood, uncertainty, and odds. Probability is usually expressed as a fraction with the denominator representing the total number of ways things can occur and the numerator representing the number of things that you are hoping will occur. Probability is always a number between 0 and 1 or between 0% and 100%. Zero means that something cannot happen (impossible) and 1 or 100% means it is sure to happen. Another way to express this is 0 ≤ P(A) ≤ 1, where A is the event. This expression is the first basic rule of probability (3).

There is also a rule that applies to two events, A and B, which are mutually exclusive, that is, the two events cannot occur at the same time. In this case, we express this as P(A or B) = P(A) + P(B). Some textbooks will use mathematical symbols for the words “and” and “or” and the expression would look like .

Although these rules of probability are extremely few and simple, they are incredibly powerful in application. In order to understand probability, you must know how many possible ways a thing can happen. For instance, if you flip a coin, there are two possible ways it can land, either heads or tails. If we want to calculate the probability of the coin landing on a head, we see that the head is one of two possible ways so the probability is or 0.5. The probabilities do not change on the second or subsequent coin tosses. This is because the events are independent. One event is not tied to a prior or future event.

As with the dice game, Craps, discussed above, every time one tosses a coin or throws the dice, the probability of that individual event occurring is the same. This concept is sometimes very difficult for people, whether engineers, politicians, or gamblers, to comprehend. If a risk analysis of an airplane is performed and a probability of the airplane breaking apart in mid-flight is calculated as 1 event in every 10,000 flights, it does not mean the event will not occur on its first flight. It has a 1 in 10,000 chance of occurring on the first and every subsequent flight. The only time the probability changes is if the assumptions on which the probability estimate was made changes.

Let us reconsider the dice game. If a thrower throws two (2) sixes, what is the probability on the next throw that double sixes will come up again? The probability is the same, 1 out of 36. The two events are independent. However, the probabilities change significantly if one is predicting the probability in advance that double sixes will be thrown twice in a row. This will be explained in another section.

The probability of winning a major lottery is 1 out of 10,000,000. A $1 ticket is purchased by our player. Our player wins. Next week our player considers playing again. What is the probability our player will win next week's lottery? It is the same, 1 out of 10,000,000. However, the probability of our player winning both lottery events is

or

This would be a very rare event, but since it is not “0,” it could occur.