PROBABILITY THEORY
What are eventsand probability?
Probabilityis a branch of mathematics that assigns a number measuring the “chance”
that some “event”—or any collection of outcomes of an experiment—will occur. It is a
quantitative description of the likely occurrence of the event, conventionally expressed
on a scale from 0 to 1. For example, a very common occurrence has a probability of close
to 1; an event that is more rare will have a probability close to 0. In more common
usage, the word “probability” means the chance that a particular event (or set of events)
will occur—all on a linear scale expressed as a percentage between 0 and 100 percent
(%). An even more detailed way of looking at probability includes the possible outcomes
of a given event along with the outcomes’ relative likelihoods and distributions.What is a sample space?
In any experiment, there are certain possible outcomes; the set of all possible out-
comes is called the sample spaceof the experiment. Each possible result is represent-
ed by one and only one point in the sample space, which is usually denoted by the let-
ter S. To each element of the sample space (or to each possible outcome) a probability
measure between 0 and 1 is assigned, with the sum of all the probability measures in
the sample space equal to 1.What are ratiosand proportions?
A ratio is the comparison of two numbers; it is most often written as a fraction or with a
“:”, as in 3/4 or 3:4 to separate the two numbers. For example, if we want to know the
ratio of dogs in a shelter that houses 24 animals to the total count, we first determine the
number of dogs (say, 10); then the ratio of dogs to animals in the shelter becomes 10/24,
or 10:24, which is also said as “10 to 24.” But there are rules to ratios. For example, order
matters when talking about ratios; therefore, the ratio 7:1 is not the same as 1:7.
A proportion is an equation with a ratio on each side, and is a statement that two
ratios are equal. For example, 1/2 4/8, or 1/2 is proportional to 4/8. In order to
“solve the proportion”—or when one of the four numbers in a proportion is
unknown—we need to use cross products to find the unknown number. For example,
to solve for xin the following: 1/4 x/8; using cross product, 4x 1 8; thus, x2.What are some simple probability events?
The probability measure of an event is sometimes defined as the ratios between the
number of outcomes. There are many simple illustrations of probability events, many
of which we are all familiar with. One of the simplest examples of probability is tossing
246 a coin, with a sample space of two outcomes: heads or tails. If a coin were completely