Section 1.4 Probability
Probability is the mathematical theory used to study uncertainty. It provides tools for the formalization and quantification of the notion of uncertainty. In particular, it deals with the chance of an event occurring. For example, if the different potential outcomes of an experiment are equally likely to occur then the probability of each outcome is taken to be one divided by the number of potential outcomes. As an illustration, consider tossing a fair coin. There are two possible outcomes — a head or a tail — and the probability of each outcome is \(1/2\text{.}\)
If you toss a fair coin 4 times, the outcomes may not necessarily be 2 heads and 2 tails. However, if you toss the same coin 4,000 times, the outcomes will be close to 2,000 heads and 2,000 tails. It is very unlikely to obtain more than 2,060 tails and it is similarly unlikely to obtain less than 1,940 tails. This is consistent with the expected theoretical probability of heads in any one toss. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when the number of repetitions is large. Statistics exploits this pattern regularity in order to make extrapolations from the observed sample to the entire population.
The theory of probability began with the study of games of chance such as poker. Today, probability is used to predict the likelihood of an earthquake, of rain, or whether you will get an “A” in this course. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of return on a client’s investments. You might use probability to decide to buy a lottery ticket or not.
Although probability is instrumental for the development of the theory of statistics, in this introductory course we will not develop the mathematical theory of probability. Instead, we will concentrate on the philosophical aspects of the theory and use computerized simulations in order to demonstrate probabilistic computations that are applied in statistical inference.
