What do the braces , P(B|A), , and P(A|B) mean in the square diagrams?
Conditions
- The visualization assumes probabilities can be represented by relative areas.
- Both A and B have positive probability for the ordinary conditionals used here.
Reasoning, step by step
- Identify the whole square as the sample space of all possibilities.
- Locate the vertical strip labeled ; its width represents the probability of A.
- Within that strip, locate the subregion labeled P(B|A); its height represents the conditional probability of B given A.
- Multiply the width by the height P(B|A) to get the area of the intersection, which is P(A and B).
- Repeat the process for the horizontal strip and its subregion P(A|B) to get the same intersection area.
Example
The right square marks as a vertical strip and P(B|A) as the lower fraction of that strip; the left square marks as a horizontal strip and P(A|B) as the left fraction of that strip.
Common misconceptions
- Confusing the conditional probability P(B|A) with the joint probability P(A and B); P(B|A) is only a fraction of the A-region, not the whole intersection area by itself.
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