Dependence (Statistical) Formula

The Formula

P(A \text{ and } B) = P(A) \times P(B|A)

When to use: Knowing A happened tells you something about B—they're connected.

Quick Example

Drawing cards without replacement: P(\text{2nd ace} \mid \text{1st was ace}) < P(\text{ace}) initially.

Notation

P(B|A) \neq P(B) indicates that A and B are dependent

What This Formula Means

When the probability of one event changes based on whether another event occurred.

Knowing A happened tells you something about B—they're connected.

Formal View

A and B are dependent if P(A \cap B) \neq P(A) \cdot P(B); then P(A \cap B) = P(A) \cdot P(B|A)

Worked Examples

Example 1

medium
A bag has 5 red and 3 blue balls. Two balls are drawn without replacement. Find P(\text{both red}) using the multiplication rule for dependent events.

Solution

  1. 1
    Event A = first ball is red: P(A) = \frac{5}{8}
  2. 2
    Event B = second ball is red, given first was red: P(B|A) = \frac{4}{7} (only 4 red left among 7)
  3. 3
    Apply multiplication rule: P(A \cap B) = P(A) \cdot P(B|A) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}
  4. 4
    Note: events are dependent because removing the first ball changes the composition of the bag

Answer

P(\text{both red}) = \frac{5}{14} \approx 0.357
Without replacement creates dependence — the outcome of the first draw changes the probabilities for the second. The general multiplication rule P(A \cap B) = P(A) \cdot P(B|A) handles both dependent and independent events.

Example 2

hard
Disease test: P(\text{disease}) = 0.05. Test positive given disease: P(+|D) = 0.90. Test positive given no disease: P(+|D^c) = 0.10. Find P(D \cap +) and P(D^c \cap +).

Common Mistakes

  • Assuming all sequential events are dependent — coin flips remain independent even if done one after another
  • Confusing dependence with causation — rain and umbrellas are statistically dependent but rain does not cause umbrellas to exist
  • Using the multiplication rule P(A) \times P(B) for dependent events, forgetting to use P(A) \times P(B|A)

Why This Formula Matters

Most real-world events are dependent—ignoring this leads to wrong calculations.

Frequently Asked Questions

What is the Dependence (Statistical) formula?

When the probability of one event changes based on whether another event occurred.

How do you use the Dependence (Statistical) formula?

Knowing A happened tells you something about B—they're connected.

What do the symbols mean in the Dependence (Statistical) formula?

P(B|A) \neq P(B) indicates that A and B are dependent

Why is the Dependence (Statistical) formula important in Math?

Most real-world events are dependent—ignoring this leads to wrong calculations.

What do students get wrong about Dependence (Statistical)?

Dependence \neq causation. Rain and umbrellas are dependent but rain doesn't cause umbrellas.

What should I learn before the Dependence (Statistical) formula?

Before studying the Dependence (Statistical) formula, you should understand: probability, independent events.

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