Joint, Marginal and Conditional Probability
Learning from data means reasoning about several uncertain quantities at once. The joint distribution holds everything, the marginal sums out what is not of interest, and the conditional is what remains once something is known.
What a learner can do afterwards
- Read a joint table and compute both marginals and a conditional from it
- Say what independence means as a statement about the joint distribution
- Show that a conditional probability and its reverse are usually different numbers
1 · Read
When you study two traits together, the joint table lists a share for every combination. Picture 40 students by gender and sport: among women the counts are 8 soccer, 8 basketball, and 4 lacrosse, and among men they are 4, 12, and 4. That full grid holds everything, and every answer you compute comes from it.
A marginal looks at one trait and sums the other away, pushing totals to the edge of the grid. Basketball draws 8 women plus 12 men, so its marginal share is 20 out of 40, which is 0.50. Change the base and you change the meaning, so always ask who is being counted.
A conditional zooms into one row or column and rescales it. Among the 20 women, 8 play soccer, so the share is 8 out of 20, which is 0.40. The base moved from 40 to 20. The same flip explains a clinic test: 9 of 12 positives are sick, which is 0.75, while 9 of 10 sick test positive, which is 0.90, so a conditional and its reverse usually differ.
Two traits are independent exactly when every joint entry equals the product of its two marginal shares. Test one cell by multiplying its row and column shares. A single cell that breaks the rule is enough to call the traits dependent.
The joint grid holds all shares, marginals sum one trait away, conditionals rescale one slice, and independence means every cell matches its product.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.