Dependence (Statistical) Formula

Dependence (statistical) is two events are statistically dependent when knowing one event occurred changes the probability of the other — formally.

The Formula

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

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

Quick Example

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

Notation

P(B∣A)≠P(B)P(B|A) \neq P(B) indicates that AA and BB are dependent

What This Formula Means

Two events are statistically dependent when knowing one event occurred changes the probability of the other — formally, P(B∣A)≠P(B)P(B|A) \neq P(B), meaning the events share information.

Knowing AA happened tells you something about BB—they're connected.

Formal View

AA and BB are dependent if P(A∩B)≠P(A)⋅P(B)P(A \cap B) \neq P(A) \cdot P(B); then P(A∩B)=P(A)⋅P(B∣A)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(both red)P(\text{both red}) using the multiplication rule for dependent events.

Answer

P(both red)=514≈0.357P(\text{both red}) = \frac{5}{14} \approx 0.357

First step

1
Event A = first ball is red: P(A)=58P(A) = \frac{5}{8}

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Example 2

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

Example 3

medium
A box has 33 defective and 77 good items. Two are picked without replacement. Find P(both defective)P(\text{both defective}).

Common Mistakes

  • Multiplying P(A)×P(B)P(A)\times P(B) for dependent events — use P(A)×P(B∣A)P(A)\times P(B|A) instead.
  • Forgetting the pool changed — after a draw without replacement, update the counts before the next probability.
  • Equating dependence with causation — linked probabilities don't prove one causes the other.

Why This Formula Matters

Dependence decides whether you can simply multiply probabilities or must use a conditional one, P(A and B)=P(A)×P(B∣A)P(A\text{ and }B)=P(A)\times P(B|A). Missing it produces wrong answers in any without-replacement or linked-event problem. Recognizing it by "Does knowing the first event occurred change the probability of the second?" — rather than by familiar numbers — is what lets a student tell it apart from independent events and conditional probability and causation in a mixed problem set.

Frequently Asked Questions

What is the Dependence (Statistical) formula?

Two events are statistically dependent when knowing one event occurred changes the probability of the other — formally, P(B∣A)≠P(B)P(B|A) \neq P(B), meaning the events share information.

How do you use the Dependence (Statistical) formula?

Knowing AA happened tells you something about BB—they're connected.

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

P(B∣A)≠P(B)P(B|A) \neq P(B) indicates that AA and BB are dependent

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

Dependence decides whether you can simply multiply probabilities or must use a conditional one, P(A and B)=P(A)×P(B∣A)P(A\text{ and }B)=P(A)\times P(B|A). Missing it produces wrong answers in any without-replacement or linked-event problem. Recognizing it by "Does knowing the first event occurred change the probability of the second?" — rather than by familiar numbers — is what lets a student tell it apart from independent events and conditional probability and causation in a mixed problem set.

What do students get wrong about Dependence (Statistical)?

The procedure for dependence (statistical) is the easy part; the trap is multiplying P(A)×P(B)P(A)\times P(B) for dependent events. Asking "Does knowing the first event occurred change the probability of the second?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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