Probability deals with experiments whose outcomes cannot be predicted with certainty. For example, tossing a coin (head or tail) or rolling a die \((1,2,3,4,5,6)\).
Probability measures how likely an event is to occur. Its value is always between \(0\) and \(1\), inclusive. An impossible event has probability \(0\), while a certain event has probability \(1\).
To quantify probabilities, we define the sample space and the events associated with an experiment.
The sample space is the set of all possible outcomes of an experiment.
Example 1. Rolling a die:
\[ S = \{1,2,3,4,5,6\} \]Example 2. Tossing two coins:
\[ S = \{HH, HT, TH, TT\} \]Example 3. Rolling two dice:
\[ \begin{aligned} S = \{ &(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),\\ &(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),\\ &(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),\\ &(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),\\ &(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),\\ &(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\} \end{aligned} \]An event is a subset of the sample space.
Example 4. Even numbers on a die:
\[ E = \{2,4,6\} \]Example 5. Two heads when tossing two coins:
\[ E = \{HH\} \]Example 6. Sum equals 4 when rolling two dice:
\[ E = \{(1,3),(2,2),(3,1)\} \]Classical probability applies when all outcomes are equally likely.
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]Example 7. Probability of rolling a 3:
\[ P(E) = \frac{1}{6} \]Example 8. Probability of rolling an even number:
\[ P(E) = \frac{3}{6} = \frac{1}{2} \]Empirical probability is based on observed data.
Suppose 30 people were surveyed about their favorite color:
| Color | Frequency |
|---|---|
| Red | 10 |
| Blue | 15 |
| Green | 5 |
Example 9. Probability a student is in grade 3:
\[ P(E) = \frac{40}{250} = 0.16 \]