STEM Calculators

Independent Versus Dependend Events

Statistics • Probability

Written by STEM Calculators Team Published Updated
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Independent vs Dependent Events

Two events A and B are independent if knowing one occurred does not change the probability of the other: P(A | B) = P(A) (equivalently P(B | A) = P(B)). This tool uses a 2×2 table to compute P(A), P(B), P(A ∩ B), conditionals, and a clear independence check.

Rows describe Variable 1 categories.

Columns describe Variable 2 categories.

Textbook-style rounding.

Event A will be one row category.

Event B will be one column category.

Enter the 2×2 counts (cells)

Male × In Favor

Male × Against

Female × In Favor

Female × Against

Cells define counts for (row category) ∩ (column category).

What this calculator checks

  • Independence test 1: P(A | B) = P(A) (when P(B) > 0).
  • Independence test 2: P(B | A) = P(B) (when P(A) > 0).
  • Equivalent test: P(A ∩ B) = P(A) · P(B) (works even if a conditional is undefined).

Mutually exclusive events have P(A ∩ B) = 0. If both P(A) > 0 and P(B) > 0, mutually exclusive events are always dependent.

Ready

Visualization

The meter shows Δ = P(A ∩ B) − P(A) · P(B). When Δ = 0, the events are independent.

Enter counts and click “Calculate”.

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Frequently Asked Questions

What does it mean for two events to be independent?

Two events are independent if the occurrence of one does not change the probability of the other. A common test is whether P(A and B) equals P(A) x P(B).

How do you check if events are independent using conditional probability?

Events are independent if P(A|B) equals P(A) and P(B|A) equals P(B), when those conditional probabilities are defined. If conditioning changes the probability, the events are dependent.

What formula should I use for dependent events?

For dependent events, use the multiplication rule P(A and B) = P(A|B) x P(B) or equivalently P(A and B) = P(B|A) x P(A). The condition tells you which probability is in the denominator.

Are independent events the same as mutually exclusive events?

No. Mutually exclusive events cannot happen together, so P(A and B) = 0, while independent events can occur together and satisfy P(A and B) = P(A) x P(B).

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