"Types of Probability (Classical, Empirical, Subjective)",

TL;DR

Probability measures how likely an event is to occur, ranging from 0 (impossible) to 1 (certain). There are three main types: Classical (based on equally likely outcomes), Empirical (based on past observations), and Subjective (based on personal belief or expert opinion). Each type is used in different situations depending on the available information.

1. The Mental Model

Think of probability as a way to quantify uncertainty. Depending on what you know about the situation, you can calculate this uncertainty using different methods: direct counting, observing what happened before, or making an informed guess.

2. The Core Material

Probability is a numerical measure of the likelihood that an event will occur. It's always a value between 0 and 1 (or 0% and 100%).

Classical Probability

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Classical probability applies when all possible outcomes of an event are equally likely. You can determine this probability before any experiment is performed.

• Formula: $P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
• Example: Rolling a fair six-sided die. The probability of rolling a '4' is 1/6 because there's one '4' and six total equally likely outcomes.

Empirical Probability

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Empirical probability (also called relative frequency probability) is based on actual observations or experiments. You calculate it after an experiment has been conducted.

• Formula: $P(\text{Event}) = \frac{\text{Number of times the event occurred}}{\text{Total number of trials}}$
• Example: If you flip a coin 100 times and it lands on heads 53 times, the empirical probability of getting heads is 53/100 = 0.53. This might differ slightly from the classical probability of 0.5 for a fair coin.

Subjective Probability

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Subjective probability is based on personal judgment, intuition, or experience. It's used when classical or empirical methods aren't feasible due to unique circumstances or a lack of data.

• Example: A doctor estimating the probability of a patient recovering from a rare disease based on their medical expertise and the patient's specific condition. A sports analyst predicting the likelihood of a team winning a championship.

Here's a diagram illustrating the decision process for choosing a type of probability:

graph TD
    A["Need to find a probability?"] --> B{"Are all outcomes equally likely?"}
    B -- Yes --> C["Use Classical Probability"]
    B -- No --> D{"Can you observe/experiment repeatedly?"}
    D -- Yes --> E["Use Empirical Probability"]
    D -- No --> F{"Is it a unique event/No data?"}
    F -- Yes --> G["Use Subjective Probability"]
    F -- No --> H["Re-evaluate situation/More advanced methods"]

3. Worked Example

Let's say you're trying to predict the outcome of a basketball game between Team A and Team B.

1. Classical Probability? Not really applicable. The outcomes (Team A wins, Team B wins) aren't equally likely. One team might be much stronger than the other. You can't just say there's a 50/50 chance without more info.

2. Empirical Probability? You could look at their past 10 games. If Team A has beaten Team B in 7 out of their last 10 matchups, the empirical probability of Team A winning this game (based on that data) is 7/10 or 0.70. This gives you a data-driven estimate.

3. Subjective Probability? Imagine it's the playoffs, and Team A's star player just got injured. Even though Team A won 7 out of 10 previous games, a sports analyst might say, "Given the star player's injury, I'd put Team A's chances of winning this specific game at only 40%." This is a subjective assessment, factoring in new, unique information that isn't captured in the historical win/loss record.

4. Key Takeaways

• Classical probability relies on counting equally likely outcomes before an event happens.
• Empirical probability is calculated from observed frequencies after an experiment or data collection.
• Subjective probability is a personal judgment based on available information, experience, or intuition.
• Choose the type of probability based on the nature of the event and the information you have available.
• All probability values range from 0 (impossible) to 1 (certain).
• Classical probability assumes an ideal scenario, like a perfectly fair coin or die.
• Empirical probability provides an estimate that gets closer to the true probability with more trials.

Common mistakes to avoid:
- Don't use classical probability if outcomes aren't truly equally likely (e.g., assuming a biased coin is 50/50).
- Don't mistake empirical probability for the absolute "true" probability, especially with a small number of trials.
- Don't present a subjective probability as an objective fact; acknowledge it's based on judgment.
- Don't confuse probability with certainty; even a high probability doesn't guarantee an outcome.

5. Now Try It

Think of an upcoming event in your life (e.g., getting a specific grade on a test, a local sports team winning, it raining tomorrow). For this event, identify which type of probability you would use to assess its likelihood (Classical, Empirical, or Subjective). Explain why that type is most appropriate and briefly describe how you would determine the probability using that method. What number (between 0 and 1) would you assign to it, and what does that number mean to you?