# 2 PROBABILITY

• It is often necessary to “guess” about the outcome of an event in order to make a decision

• Probability is a measure that is associated with how certain we are of outcomes of a particular experiment

• The sample space, denoted with $$S$$ is the collection of all possible outcomes of the random experiment

• Combination (subset) of the outcomes is an event. Events are usually denoted with upper case letters like $$A$$, $$B$$, etc.

How we calculate the probability of an event $$A$$ when the outcomes are equally likely?

\begin{align} P(A)&=\frac{m}{n} \\ \\ m&=\text{number of outcomes that satisfy event A} \\ \\ n&=\text{number of all possible outcomes} \\ \tag{2.1} \end{align}

• Probability of any event is always non-negative number between $$0$$ amd $$1$$, i.e. $$0\leq P(A) \leq 1$$

• $$P(A)=0$$ meansthat event $$A$$ will not occur (impossible event), while $$P(A)=1$$ means that event $$A$$ will occur with $$100\%$$ certainty

How we calculate the probability of an event $$A$$ when the outcomes are not equally likely?

\begin{align} P(A)&=\frac{m_n}{n} \\ \\ m_n&=\text{number of outcomes that satisfy event A in n repetitions} \\ \\ n&=\text{number of repetitions (trials) of an experiment} \\ \tag{2.2} \end{align}

The law of large numbers states: as the number of repetitions of an experiment is increased, the

relative frequency tends to become closer and closer to the theoretical (true) probability