What is probability?

Probability quantifies how likely it is that an event will occur. Generally speaking, there are many instances in real life where we need to make predictions about how something will turn out. The outcome of an event may be known to us or unknown to us. When this happens, we say that there is a chance that the event will happen or not.

Mathematical formulation

In mathematical terms, probability is defined as the ratio of the number of ways that an event can take place to the total number of outcomes of an event. It is depicted as follows:

According to this equation, probability is determined by dividing the total number of possible outcomes of an event by the number of ways that a favorable event can occur. For example, the probability of rolling a two (or any number) on a six-sided (fair) die would be $\frac{1}{6}$.

Note: The probability of an event can only be between 0 and 1, and can also be written as a percentage.

Types

There are three types of probability:

• Theoretical probability

• Experimental probability

• Axiomatic probability

Theoretical probability

Theoretical probability depends on the likelihood of something occurring. It is based on what is expected to happen in an experiment that is not being conducted. It is the number of favorable outcomes divided by the total number of outcomes.

Experimental probability

Experimental probability is defined as the number of possible outcomes divided by the total number of trials. For example, we have only two possible outcomes when flipping a coin: heads or tails. The total number of trials is determined by the number of coin flips. If we toss the coin 20 times and 3 of them land on heads, the experimental probability is 3/20.

Axiomatic probability

Axiomatic probability establishes a set of axioms that apply to all probability approaches. The chances of events occurring and not occurring can be quantified using this probability. It is the probability that an event will occur based on the occurrence of a previous event.

Example

Let's assume we want to find the probability of getting a number less than 3 when a dice is rolled. Using the formula given above, we first find the number of ways this can occur, $n(A)$, and the total number of outcomes, $n(T)$.

We calculate this as follows:

Key terms

The terms listed below can help us better understand probability concepts.

Trial: This refers to the multiple attempts made during the course of an experiment. For example, tossing a coin is a trial.

Event: This is a trial with a distinctly defined outcome. An event is, for example, getting a tail when tossing a coin.

Sample space: This comprises all the possible outcomes of an experiment. Tossing a coin, for example, has a sample space of head and tail.

Outcome: This is the final outcome of a trial. In the process of a coin toss, there are two clear outcomes: a person may either win or lose.

Random event: A random event is one that cannot be easily predicted. The probability value for such events is extremely low.

Exhaustive events: These occur when the set of all outcomes of an experiment equals the sample space.

Mutually exclusive events: These events are those that cannot occur at the same time. A coin toss can result in either head or tails, but neither outcome can occur at the same time.