Textbook
Events $A$ and $B$ are independent if $P(A\cap B)=P(A)P(B)$. For $P(B)>0$, this is equivalent to $P(A\mid B)=P(A)$: knowing $B$ occurred does not change the chance of $A$. The product definition avoids division and also covers zero-probability events.
For finite-valued random variables $X$ and $Y$, independence means $P(X=x,Y=y)=P(X=x)P(Y=y)$ for every possible pair. In general, the corresponding equality must hold for events that the variables lie in admissible sets. Notation includes $X\perp Y$ and $X\perp\!\!\!\perp Y$.
Independent coin tosses are an assumption of a model. Separate spreadsheet columns do not establish independence. Disjointness is different: if two positive-probability events cannot occur together, their intersection has probability zero while the product of their probabilities is positive.
With finite second moments, independence implies zero covariance; the converse fails. A single numerical measure of association cannot replace the condition for all events or all pairs of values.
The condition is symmetric because multiplying probabilities does not depend on event order. With positive probabilities, it implies both unchanged $P(A\mid B)$ and unchanged $P(B\mid A)$. For finite-valued variables, every value pair must be checked, including impossible pairs. If both individual values have positive probability, independence requires a positive chance that they occur together. Checking one convenient pair generally cannot replace checking the whole joint distribution.
Definitions and notation: Mathematics for Computer Science; Mathematics for Machine Learning.
Related concepts: Conditional probability: reweighting after a condition, Covariance: deviations together.
In plain language
Under independence, information about the first coin does not change the probability of the second coin’s result. This concerns the joint probability model rather than anyone’s personal surprise. If the second result simply copies the first, two columns contain the same information. Two windows can show one film; moving the windows apart on a screen does not make them independent.
Example
The constructed model has two independent fair coin tosses. In outcome pairs, H means heads and T means tails. Event $A$ means heads first and $B$ heads second. The comparison then uses a copied result.
Statistical independence
Joint outcomes
Four pairs have equal weight 1/4.
Step 1 of 4: Joint outcomes
Read all steps
Joint outcomes
Four pairs have equal weight 1/4.
- Pairs
- HHHTTHTT
Both heads
The intersection contains one pair, with probability 1/4.
- A and B
- HHincludedHTexcludedTHexcludedTTexcluded
Second is heads
Among HH and TH, half have heads first.
- Remaining pairs
- HHincludedTH
- Conditional probability
- 1/2included
A copied result
A different model has only HH and TT, each of weight 1/2: intersection exceeds the product.
- Pairs in the new model
- HHincludedTT
- Intersection; product
- 1/2excluded1/4current
In the independent model, each event occupies two of four equally likely pairs, while joint heads occupies one. Conditioning on heads second leaves two pairs, only one of which has heads first. Under copying, the separate heads probabilities remain one half, but the joint probability becomes one half instead of one quarter. Marginal distributions alone are therefore insufficient: identical separate probabilities can coexist with independence or complete dependence.
Independence is either an assumption or a consequence established through joint probabilities. A table layout is not a proof.