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