# Probability

Probability theory › Probability

Probability is a measure quantifying the likelihood that events will occur.

Probability quantifies as a number between 0 and 1, where, loosely speaking,\[note 1] 0 indicates impossibility and 1 indicates certainty.

The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).

These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in such areas of study as mathematics, statistics, finance, gambling, science (in particular physics), artificial intelligence/machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.

[https://revisionmaths.com/gcse-maths-revision/statistics-handling-data/probability](https://revisionmaths.com/gcse-maths-revision/statistics-handling-data/probability)

## Single Events

Probability is the frequency at which an event occurs out of a greater number of outcomes.

Probability is the likelihood or chance of an event occurring.

Probability is the number of ways of achieving success divided by the number of total possible outcomes.

Experimental probability and the importance of basing this on a large trial sample.

Expressing probability as fractions and percentages based on the ratio of the number ways an outcome can happen and the total number of outcomes.

The probability of something which is certain to happen is 1. The probability of something which is impossible to happen is 0. The probability of something not happening is 1 minus the probability that it will happen.

## Multiple Events

Independent and Dependent Events

Suppose now we consider the probability of 2 events happening. For example, we might throw 2 dice and consider the probability that both are 6's.

We call two events _independent_ if the outcome of one of the events doesn't affect the outcome of another.

For example, if we throw two dice, the probability of getting a 6 on the second die is the same, no matter what we get with the first one- it's still 1/6.

When the probability of one event depends on another, the events are _dependent_.

Suppose we have a bag containing 2 red and 2 blue balls. If we pick 2 balls out of the bag, the probability that the second is blue depends upon what the colour of the first ball picked was. If the first ball was blue, there will be 1 blue and 2 red balls in the bag when we pick the second ball. So the probability of getting a blue is 1/3. However, if the first ball was red, there will be 1 red and 2 blue balls left so the probability the second ball is blue is 2/3.

## Probability Trees and Spaces

The way of representing two or more events is on a probability space or tree, where all possible outcomes are plotted.

## The AND and OR rules (HIGHER TIER)

In the above example, the probability of picking a red first is 1/3 and a yellow second is 1/2. The probability that a red AND then a yellow will be picked is 1/3 × 1/2 = 1/6 (this is shown at the end of the branch). The rule is:

If two events are independent, then the _probability of both events occurring_ is found by _multiplying their individual probabilities_.

The probability of picking a red OR yellow first is 1/3 + 1/3 = 2/3. The rule is:

If two events can't occur simultaneously, then the _probability of either event occurring_ is found by _summing their individual probabilities_.

On a probability tree, when moving from left to right we multiply and when moving down we add.

[https://revisionmaths.com/gcse-maths-revision/statistics-handling-data](https://revisionmaths.com/gcse-maths-revision/statistics-handling-data)

[https://en.wikipedia.org/wiki/Probability](https://en.wikipedia.org/wiki/Probability)
