Classical Probability

Classical probability (also known as theoretical or a priori probability) is the oldest approach to calculating probability. It was developed in the 17th century by mathematicians like Blaise Pascal and Pierre de Fermat to solve problems related to games of chance (such as rolling dice or drawing cards).

The entire foundation of classical probability rests on one crucial assumption: all possible outcomes of an experiment are equally likely. If this assumption holds true, we don't need to perform the experiment hundreds of times to figure out the probability; we can simply calculate it using logic and mathematics.

1. The Classical Definition of Probability

If a random experiment has $n(S)$ mutually exclusive, exhaustive, and equally likely outcomes, and if $n(E)$ of these outcomes are favorable to an event $E$, then the classical probability of event $E$ occurring is defined as:

$$ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{n(E)}{n(S)} $$

Key Terms:

• $S$ (Sample Space): The set of all possible outcomes.
• $E$ (Event): The specific outcome or set of outcomes we are interested in.
• $n(S)$: Total count of possible outcomes.
• $n(E)$: Total count of favorable outcomes.

Example: Drawing a Card What is the probability of drawing a King from a standard, well-shuffled deck of 52 cards? * Total possible outcomes $n(S) = 52$ (since there are 52 cards). * Favorable outcomes $n(E) = 4$ (since there are 4 Kings in a deck: Hearts, Diamonds, Clubs, Spades). * $P(\text{King}) = \frac{4}{52} = \frac{1}{13}$.

2. The Probability Value Range

Because the number of favorable outcomes $n(E)$ can never be less than zero, and it can never be greater than the total number of possible outcomes $n(S)$, the probability of any event $E$ must always fall between 0 and 1, inclusive.

$$ 0 \le P(E) \le 1 $$

• If you calculate a probability and get a negative number (e.g., -0.5) or a number greater than 1 (e.g., 1.2), you have made a mathematical error.
• Probabilities can be expressed as fractions ($\frac{1}{2}$), decimals ($0.5$), or percentages ($50\%$).

3. Certain and Impossible Events

The bounds of the probability range (0 and 1) represent two extreme types of events.

The Impossible Event: $P(E) = 0$

An event that has absolutely no chance of occurring is called an impossible event. Its subset contains zero sample points ($n(E) = 0$). * Example: Rolling a "7" on a standard six-sided die. Since the faces only go up to 6, $n(\text{Rolling 7}) = 0$. Therefore, $P = \frac{0}{6} = 0$.

The Certain (Sure) Event: $P(E) = 1$

An event that is guaranteed to occur is called a certain event. In this case, every possible outcome is a favorable outcome, meaning $n(E) = n(S)$. * Example: Rolling a number less than 10 on a standard six-sided die. Since all faces (1, 2, 3, 4, 5, 6) are less than 10, $n(E) = 6$. Therefore, $P = \frac{6}{6} = 1$.

4. The Complement Rule

As we learned in previous topics, the complement of an event $E$ (denoted as $E'$ or $\bar{E}$) consists of all outcomes where event $E$ does not occur.

Because an event and its complement are mutually exclusive and exhaustive, their probabilities must add up to exactly 1 (which represents the entire sample space).

$$ P(E) + P(E') = 1 $$ $$ P(E') = 1 - P(E) $$

Why is this useful? Sometimes, calculating the probability of an event not happening is much mathematically simpler than calculating the probability of it happening.

• Example: You roll two dice. What is the probability of NOT rolling double sixes?
  • It's tedious to count all the ways to not roll double sixes.
  • Instead, find $P(\text{Double Sixes})$. There is only 1 way to roll a (6, 6) out of 36 possible outcomes. So, $P(\text{Double Sixes}) = \frac{1}{36}$.
  • Using the complement rule: $P(\text{NOT Double Sixes}) = 1 - \frac{1}{36} = \frac{35}{36}$.

5. Limitations of Classical Probability

While classical probability is elegant and foundational, it has strict limitations in the real world:

1. The "Equally Likely" Requirement: The formula $P = \frac{n(E)}{n(S)}$ completely breaks down if the outcomes are not equally likely. If a coin is weighted (biased) to land on Heads 70% of the time, you cannot use classical probability. You must use empirical (experimental) probability.
2. Infinite Sample Spaces: Classical probability requires you to be able to count the number of outcomes. It cannot handle situations where there are infinitely many outcomes (continuous variables), such as measuring the exact height of a tree or the exact time a radioactive atom decays.

6. Basic Probability Problem Solving Strategies

When approaching classical probability problems, follow these steps:

1. Identify the Experiment: What physical action is taking place? (e.g., drawing 2 cards, rolling 3 dice).
2. Determine $n(S)$: Calculate the total number of possible outcomes. You may need to use combinatorics (permutations and combinations) or multiplication principles for this.
3. Define the Event $E$: What specific outcome are you looking for?
4. Determine $n(E)$: Count how many ways your specific event can happen.
5. Calculate the Ratio: Divide $n(E)$ by $n(S)$ and simplify the fraction.