Joint Probability: $P(A \cap B)$. The probability of inherently satisfying BOTH conditions at once (the specific cell).
Marginal Probability: The probability of satisfying ONE condition, entirely ignoring the other (the column or row totals / margins).
Conditional Probability: $P(A | B)$. "Given that we are ALREADY in column B, what is the chance of finding row A within it?" We shrink our entire denominator universe to just column B.
Testing Independence: Events are Independent if $P(B | A) == P(B)$. Meaning, knowing event A happened provides absolutely ZERO new mathematical information about event B.
Tags
CategoricalProbabilityIndependenceConditioning
GENERAL MULTIPLICATION RULE
$P(A \cap B) = P(A) \times P(B | A)$
Plays Sport ($S$)
No Sport ($S^c$)
Marginal Row Total
Takes AP ($A$)
40
20
60
No AP ($A^c$)
30
110
140
Marginal Col Total
70
130
200
Data Matrix Presets
Probability Queries
CALCULATION PATH
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INDEPENDENCE TEST
$P(\text{Sport}) =$--%
$P(\text{Sport} | \text{AP}) =$--%
Independence Check: Does knowing a student takes an AP class change the math probability that they also play a sport? If the percentages are different, the events are mathematically Dependent (associated).