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