Probability helps scientists understand why something happens or the likelihood that something will happen. This is important in statistics, where a large number of possible outcomes may occur. The theory of probability is particularly useful when the number of possible outcomes is so large, it is impractical to count all possibilities.

In probability, we talk about 'outcomes'. For example, when a coin is tossed, there are two possible outcomes: heads or tails. A simple probability can be calculated if the likelihood of any of these outcomes is equal in each case. We say that an event E has n(E) outcomes with equal probability of occurring.

The sample space of event E is the number of all possible outcomes, .

A sample space is the set of all possible outcomes for a particular event, where each outcome has equal likelihood of occurring.

For example, a single toss of a coin has a sample space of 2. This is written as S = {H, T}, where H is the outcome head, and T the outcome tail. The number of elements of the sample set is written as n(S).

The sample space for two coins being tossed is S = {HH, HT, TH, TT}, where HH is the event two heads, HT heads then tails, and so on.

The probability of event E occurring is P(E) = ${n(E)}/{n(S)}$

where P(E) is the probability that event E will occur, n(E) the number of positive outcomes for the event, and n(S) the number of all outcomes (positive and negative).

The total number of outomes for an event occurring cannot be greater than the total of all possible outcomes. Therefore, the maximum proability of an event occurring is 1. Likewise, if there are no possible outcomes for a particular event, the probability is 0. P(E) therefore has limits of 0 and 1:

0 ≤ P(E) ≤ 1

Probability may be written as a fraction (1/4), a decimal (0.25), or as a percentage (25%).

The event 'tails' is the complement to the event 'heads' for a single toss of coin. The complement to P(E) is written as P(E'). The sum of these probabilities equals 1.

P(E) + P(E′) = 1

Therefore, the probability of an event occurring is one minus the probability of its not occurring:

P(E) = 1 - P(E not occurring) = 1 - P(E′)

In this example, these are the number of mistakes a student made in ten maths tests:

4 | 2 | 3 | 4 | 7 | 6 | 4 | 2 | 1 | 0 | 5 |

The mean is the total of all the values, divided by the number of values. If the data is given with frequencies, the values need to be multiplied by their frequency, and the total is divided the the sum of the freqencies.

Value | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Total |

Frequency | 1 | 1 | 2 | 1 | 3 | 1 | 1 | 1 | 11 |

Value x Frequency | 0 | 1 | 4 | 3 | 12 | 5 | 6 | 7 | 38 |

The mean is ${38}/{11} = 3.45$

This can be written more formally as: Mean = μ = ${∑(v⋅f)}/{∑f}$

The median is the data point which is midway in the range of values. There are an equal number of values above as below the median. To determine the median, it is necessary to arrange the data in ascending or descending order.

0 | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 6 | 7 |

The median is 4. It is the middle value, which is easy to find when there is an odd number of values.

If there is an even number of values, the median is between the two middle values.

The mode is the most frequent value from a sample set.

The modal number of mistakes is 4, because it occurred 3 times. The median in an evenly distributed and spread set of data can be close to the mode, but it is not necessarily so. In this case, the median is 3.5.

Discrete data can only take specific values. For example, shoe sizes are either 8 or 8$1/2$, not 8.234....

Discrete data are usually counted and are best presented in a bar chart.

Continuous data can take any value. For example, heights can be reported as accurately as the means of measurement allows: 165 cm, or 165.2cm, or 165.24cm, etc.

Continuous data are not usually counted and are best presented in a line chart.

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