Understanding Compound Probability

Compound probability involves finding the likelihood of two or more events occurring. To solve these problems, we use two main rules: the Multiplication Rule for "AND" scenarios and the Addition Rule for "OR" scenarios.

The Multiplication Rule ($P(A \cap B)$)

The multiplication rule is used when you want to find the probability of event A and event B happening.

Independent vs. Dependent Events

• Independent Events: The outcome of the first event does not affect the second. Formula: $P(A \text{ and } B) = P(A) \times P(B)$
• Dependent Events: The outcome of the first event changes the probability of the second (e.g., drawing without replacement). Formula: $P(A \text{ and } B) = P(A) \times P(B|A)$ (Where $P(B|A)$ is the probability of B given that A has occurred).

Example: Drawing without replacement A bag has 5 red and 3 blue balls. Two are drawn without replacement. Find $P(\text{both red})$.

1. The probability of drawing a red ball first is $P(\text{1st red}) = \frac{5}{8}$.
2. Since the ball is not replaced, there are now 4 red balls and 7 total balls left. So, $P(\text{2nd red} | \text{1st red}) = \frac{4}{7}$.
3. Multiply them together: $P(\text{both red}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$

The Addition Rule ($P(A \cup B)$)

The addition rule is used when you want to find the probability of event A or event B happening (or both).

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

We subtract $P(A \cap B)$ (the intersection) because simply adding $P(A)$ and $P(B)$ double-counts the scenario where both events happen at the same time.

Example: Finding the Union Given $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cap B) = 0.15$. Find $P(A \cup B)$.

$P(A \cup B) = 0.4 + 0.5 - 0.15 = 0.75$

Testing for Independence

You can mathematically prove whether two events are independent. Events $A$ and $B$ are strictly independent if and only if their intersection equals the product of their individual probabilities:

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

Example: Are A and B independent? Using the values from the previous example: $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cap B) = 0.15$.

1. Calculate $P(A) \times P(B)$: $0.4 \times 0.5 = 0.20$
2. Compare to $P(A \cap B)$: $0.20 \neq 0.15$

Because the products are not equal, events $A$ and $B$ are dependent.