Test Whether Two Attributes Are Independent
Compare a conditional probability with its unconditional probability.
Supplied materials
- Paper or a note app.
- Fictional uniform box: red circle 4, red noncircle 8, blue circle 4, blue noncircle 8; total 24 cards.
Learn the idea
Two events are independent when learning one does not change the probability of the other. Here red has probability 12/24 = 1/2 before any shape information. Among the eight circle cards, four are red, so P(red given circle) is also 1/2.
Matching these probabilities verifies independence for this exact supplied uniform model, not a claim about unrelated real-world attributes. Another equivalent check is P(red and circle) = P(red) × P(circle). A changed cell count can change the conclusion even if the overall red total stays the same.
- Find P(red) = 12/24 and P(red given circle) = 4/8.
- Compare them and check 4/24 = (12/24)(8/24).
- Test the altered counts in the question without assuming the conclusion stays the same.
See the method in action
P(red) = 1/2 and P(red given circle) = 1/2, so red and circle are independent here. The joint check gives 1/6 = (1/2)(1/3).
Check your reasoning
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See the answer and explanation
No, because 6/8 differs from 12/24
P(red given circle) = 6/(6 + 2) = 3/4, while P(red) = 12/24 = 1/2. Knowing circle changes the red probability, so the events are dependent in this model.
Take it further
Use counts 3,9,3,9 instead. P(red given circle) = 3/6 = 1/2 equals P(red) = 12/24 = 1/2, so independence is restored.
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