If more people took the time to really understand probability, casinos would go out of business. Fortunately for those who operate games of chance, most customers do not have a great appreciation for how this function of Mathematics operates. It is not as complex of a concept as one might expect.
It is easiest to start off with these. All probabilities can be expressed as one of the real numbers in the set from 0 to 1. 0 is the lowest probability. An event which has no chance of occurring has a probability of 0, and an event that will happen with certainty has a probability of 1. There are few events in the world that can be accurately assigned one of these probabilities, but there are many events that come very close to these.
Expression of Probabilities
The normal probability values that are assigned fall between these two values. Probabilities are typically expressed as a decimal or a fraction, but they can also be expressed as a ratio, such as 1 in a million. A probability is simply the ratio of successful outcomes to all possible outcomes. For instance, if there are four balls and only one is red, the chance of randomly grabbing a red ball is 0.25, or ¼, or 1 in 4.
The fundamental concept that many do not understand about probabilistic events is that if they are truly independent, then the previous outcomes have no impact on the current trial. For example, even if a quarter has been flipped 10 times and has come out heads all 10 times, the chance of getting heads during the next flip is still only ½, assuming that the coin is fairly balanced and has a heads and tails side. Many have the mistaken impression that there is a greater chance of obtaining a heads on a flip because of the result of the previous 10 trials, but this is not true.
Series of Probabilities
Part of the reason for this way of thinking is because the probability of flipping heads 11 times in a row is very low, being only 1 in 2048. However, this is also the probability of flipping heads 6 times and then tails 5 times, or the probability of any one of the other 2046 combinations. When considering the probability of several different events, the probability of any given overall outcome is the product of all of the individual outcomes, which in this case is ½ times 11 or 1/2048.
Probability theory is an interesting area of mathematics. Some people earn their living calculating probabilities of events, but the basics are simple enough that anyone can understand. If you remember nothing else, however, remember that the lottery is simply a tax on those who are bad at math.