Probability is a concept we encounter daily, whether checking the weather forecast or contemplating lottery odds. It can be calculated mathematically, allowing us to quantify the likelihood of various events. In probability notation, we denote the probability of an event as P(event). An event can be any occurrence, such as rain or flipping heads on a coin.
There are two primary types of probability: theoretical and empirical. Theoretical probability is based on the possible outcomes before any events occur. For instance, when flipping a coin, the theoretical probability of landing heads is calculated as:
$$P(\text{heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{1}{2}$$
In contrast, empirical probability is derived from actual experiments or observations. If we flip a coin three times and get heads twice, the empirical probability is:
$$P(\text{heads}) = \frac{\text{Number of times heads occurred}}{\text{Total trials}} = \frac{2}{3}$$
While theoretical probability provides a baseline expectation, empirical probability reflects real-world outcomes, which may differ due to sample size and randomness. For example, when rolling a six-sided die, the probability of rolling a number greater than 3 can be calculated as:
$$P(\text{number} > 3) = \frac{3}{6} = \frac{1}{2} = 0.5$$
When using data from multiple rolls, such as rolling the die 10 times, if we find that 8 rolls resulted in a number greater than 3, the empirical probability would be:
$$P(\text{number} > 3) = \frac{8}{10} = \frac{4}{5} = 0.8$$
The difference between theoretical and empirical probabilities often arises from the number of trials conducted. A larger sample size tends to yield results closer to theoretical expectations. In probability studies, all possible outcomes of an event can be represented as a sample space, denoted in set notation. For example, the sample space for flipping a coin is:
$$S = \{ \text{heads, tails} \}$$
Understanding these foundational concepts of probability equips you to analyze and interpret data effectively, whether in academic settings or everyday decision-making.
