### Chapter 3. Probability: Probability

### Definition of Probability

In everyday language, probabilities are often described as percentages. In statistics, however, all probabilities are expressed as a number between #0# and #1#.

"The probability that a fair coin comes up head is #50\%#" translates to "the probability that a fair coin comes up head is #0.5#" in statistics.

A probability of #0# or a probability of #1# follow these rules:

- An event that has no chance of occurring has a probability of #0#
- e.g. the probability that someone is born on February 30th is #0#.

- An event that is certain to occur has a probability of #1#
- e.g. if today is Thursday, then the probability that tomorrow is Friday is #1#.

The greater the probability of an event, the more likely it is that the event will occur.

Probability

**Definition**

**Probability** is the likelihood that an event will occur, quantified as a number between #0# and #1#.

If the probability of an event #A# occurring is #p#, this is written as: #\mathbb{P}(A)=p#.

For an experiment in which several different outcomes are possible, the probability of any specific outcome is defined as a fraction or proportion of all the possible outcomes.

**Formula**

\[\mathbb{P}(A) = \cfrac{\text{number of outcomes classified as }A}{\text{total number of possible outcomes}}\]

Rules

- The probability of any event A, #\mathbb{P}(A)#, is always greater than or equal to #0# and less than

or equal to #1#:

\[0\leq \mathbb{P}(A) \leq 1\] - The sum of the probabilities of all possible outcomes must equal #1#. That is, if the sample space is #\Omega#, then

\[\sum_{\text{all }x \text{ in }\Omega}\mathbb{P}(x)=1\]

\[\Omega =\{\text{H, T}\}\]

The outcome belonging to event #A# is:

\[A=\text{'the coin comes up Heads'}=\{\text{H}\}\]

Thus, the probability of #A# occurring is:

\[\mathbb{P}(A)=\cfrac{\text{number of outcomes classified as 'the coin comes up Heads'}}{\text{total number of possible outcomes}}=\cfrac{1}{2}=0.5\]

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