Tree Diagrams Without Replacement
Draw tree diagrams for two or three stages where the first outcome changes the second, multiply along a branch, and add the routes that fit the question.
What a learner can do afterwards
- Adjust the second set of branch probabilities when an item is not replaced
- Multiply along a branch and add the branches that satisfy the condition
- Check that the branches at each stage add to 1
1 · Read
When you deal cards and keep each card out, the tree still follows the branch rules for combined events, but the second stage changes. The two draws are not independent: the first result changes what is left, so it changes the second-stage probabilities. Each branch carries its probability, and the branches leaving one point always add to 1. When the item is not put back, the second stage must be rebuilt from what is left.
A bag holds cards 1 to 7: four odd, three even. First draw: P(even) = 3/7. Set one even card aside and rebuild: 6 cards left with 2 even, so P(even second) = 2/6 = 1/3. Both totals dropped by one because the removed card is gone.
A route's chance is the product along its branches. Even then even gives 3/7 times 2/6 = 6/42 = 1/7. Odd then even gives 4/7 times 3/6 = 12/42 = 2/7. When several routes fit the question, add them: one odd and one even in either order is 2/7 plus 2/7 = 4/7.
At every branching point, check the probabilities add to 1. If they do not, a fraction was built from the old totals. Rebuild each stage from the cards that are actually left.
Rebuild the second stage from what is left, multiply along a route, add the routes that fit.
The rewritten reading was written for this dataset.
2 · Watch
En anglaisTake it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
14 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.