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), meaning the events share information.

The Formula

P(A and B)=P(A)×P(B∣A)

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

Quick Example

Drawing cards without replacement: P(2nd ace∣1st was ace)<P(ace) initially.

Notation

P(B∣A)≠P(B) indicates that A and B 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), meaning the events share information.

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

Formal View

A and B are dependent if P(A∩B)≠P(A)⋅P(B); then P(A∩B)=P(A)⋅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) using the multiplication rule for dependent events.

Answer

P(both red)=514≈0.357

First step

1
Event A = first ball is red: P(A)=58

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Example 2

hard
Disease test: P(disease)=0.05. Test positive given disease: P(+∣D)=0.90. Test positive given no disease: P(+∣Dc)=0.10. Find P(D∩+) and P(Dc∩+).

Example 3

medium
A box has 3 defective and 7 good items. Two are picked without replacement. Find P(both defective).

Common Mistakes

  • Multiplying P(A)×P(B) for dependent events — use P(A)×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). 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), meaning the events share information.

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)≠P(B) indicates that A and B 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). 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) 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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