Differentiate between classical, empirical, and subjective probability. Provide examples for each type.
The Correct Answer and Explanation is :
Classical Probability:
Classical probability, also known as theoretical probability, refers to situations where all outcomes are equally likely. It is based on the assumption that every outcome in a given sample space has an equal chance of occurring. This type of probability is typically used in games of chance, such as rolling dice, flipping coins, or drawing cards from a well-shuffled deck.
Example:
When rolling a fair six-sided die, the classical probability of rolling a three is calculated as follows:
[
P(\text{rolling a 3}) = \frac{1}{6}
]
since there are six equally likely outcomes (1, 2, 3, 4, 5, 6).
Empirical Probability:
Empirical probability, also called experimental probability, is based on observed data or experiments. It is the ratio of the number of successful outcomes to the total number of trials or observations. Empirical probability is useful when the theoretical model is difficult to apply, and actual data is available for analysis.
Example:
Suppose a coin is flipped 100 times, and it lands on heads 52 times. The empirical probability of getting heads is:
[
P(\text{heads}) = \frac{52}{100} = 0.52
]
This probability is based on actual observed results.
Subjective Probability:
Subjective probability is based on an individual’s personal judgment or belief about the likelihood of an event occurring. This type of probability does not rely on data or mathematical reasoning but instead is influenced by personal experience, intuition, or opinion. It is commonly used in situations where empirical data is scarce or nonexistent.
Example:
A person might believe that there is a 70% chance it will rain tomorrow, based on their knowledge of weather patterns or a gut feeling, even if there is no hard data or model supporting this probability.
Conclusion:
In summary, classical probability relies on equal likelihood of outcomes, empirical probability is based on observed frequencies, and subjective probability depends on personal beliefs or judgment. Each type has its application depending on the context of the problem and the available information.