Introduction to Probability Theory

1. The Multiplication Rule ("And")

When we want to find the probability of two events occurring together or in succession (Event A and Event B), we use the multiplication rule.

If the events are independent—meaning the outcome of the first does not change the likelihood of the second—the formula is:

$$P(A \cap B) = P(A) \cdot P(B)$$

Example: If you flip a coin and roll a die, what is the probability of getting "Heads" and a "6"? Since the coin flip doesn't affect the die roll, these are independent.

$$P(\text{Heads} \cap 6) = P(\text{Heads}) \cdot P(6) = \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12}$$

2. The Addition Rule ("Or")

When we want to find the probability of either Event A or Event B occurring, we use the addition rule. However, we must be careful not to "double-count" the instances where both events happen at the same time.

The general formula is:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

If the events are mutually exclusive (meaning they cannot happen at the same time, like rolling a 3 and a 5 on a single die), then $P(A \cap B) = 0$, and the formula simplifies to:

$$P(A \cup B) = P(A) + P(B)$$

Conditional Probability and Dependence

A critical concept in advanced probability is conditional probability, which is the probability of an event occurring given that another event has already occurred. This is written as $P(A|B)$, read as "the probability of A given B."

When events are dependent, the occurrence of Event B changes the probability of Event A. This often happens in "sampling without replacement."

Worked Example: The Marble Bag Imagine a bag contains 3 red marbles and 7 blue marbles (10 total). You draw two marbles one after another without putting the first one back. What is the probability that both are red?

1. First Draw: The probability of drawing a red marble is $P(R_1) = \frac{3}{10}$.
2. Second Draw: Since we didn't replace the first marble, there are now only 2 red marbles left and 9 total marbles. The probability of drawing a second red marble given the first was red is $P(R_2|R_1) = \frac{2}{9}$.
3. Combined Probability:

$$\begin{aligned} P(R_1 \cap R_2) &= P(R_1) \cdot P(R_2|R_1) \\ &= \frac{3}{10} \cdot \frac{2}{9} \\ &= \frac{6}{90} = \frac{1}{15} \end{aligned}$$

Key Takeaway: Always ask yourself: "Does the first action change the environment for the second action?" If yes, you are dealing with dependent events and must use conditional probability.

Summary Table of Rules

To help with your studies, refer to this quick summary of the core principles discussed:

B) = \frac{P(A \cap B)}{P(B)}$ |