Probability trees for a two-step game
A probability tree multiplies along a path and adds across paths. Two hearts from a deck show why the second chance shrinks.
A probability tree is a picture of steps. Each branch is one way the step can go, labeled with its chance. You multiply along a path because every step on that path has to happen. You add across paths when any one of those paths counts as success, and only after you have checked that the paths do not overlap.
This is for adults 18 and older. The deck example is exact. It is not a staking plan. Laws differ by place.
Two hearts, drawn in order
A standard deck has 52 cards and 13 hearts. The chance the first card is a heart is 13/52, which is 1/4. Given that it was a heart, 12 hearts remain in 51 cards, so the second-step chance is 12/51. Multiply along the path: 13/52 times 12/51 equals 156/2652, which simplifies to 1/17.
Check the simplification. 13/52 is 1/4, and 12/51 is 4/17, and one quarter of 4/17 is 1/17. The same product is 156/2652, and 156 times 17 is 2652. The tree did not add new information. It stopped you from using 13/52 twice, which would pretend the first heart was replaced.
That second branch is conditional. The tree makes the condition visible. A story problem hides it inside the word "another."
Adding the paths that count
Suppose you care about the first card being an ace or a king, and you stop there. Those are two branches. An ace has chance 4/52. A king has chance 4/52. They cannot happen together, so the combined chance is 8/52, which is 2/13. Adding is legal because the branches are mutually exclusive.
If you instead want "at least one heart in two cards," you have several paths: heart then non-heart, non-heart then heart, and heart then heart. Add those three products. Do not also add a path you already included inside another. Listing the leaves of the tree, once each, is the entire discipline. The chance of no heart at all is often easier: 39/52 times 38/51. Subtract that from 1 and you have at least one heart. Both methods should match. If they do not, a branch was dropped or doubled.
Where trees get abused
A tree for a parlay looks neat and still lies if the branches are not really separate. Two bets in the same game share a path the drawing does not show. Parlay math fails when people multiply dependent legs as if they were hearts leaving a deck. The picture is only as honest as the chances on the twigs.
A tree also does not shrink variance. A path with probability 1/17 will fail sixteen times for every time it hits, in the long-run average, and the short run will not look that tidy. The diagram is a price input. It is not a schedule.
Draw it, then put it away
Sketch three steps at most unless you are willing to check the arithmetic twice. Label every branch with a fraction that sums to 1 at each split. Hearts and non-hearts on the first card, 13/52 and 39/52, already use the whole deck. If your branches sum to 90 percent, the missing 10 percent is a hole in the question.
Then stop. A correct tree will not make a negative house edge fair, and it will not justify a stake from bill money. If drawing trees is how you talk yourself into another session, close the notebook. Local help is a better next branch than another bet.
Frequently asked questions
Do I multiply or add on a tree? Multiply along a single path. Add across paths that each count as the result you want, and only if those paths do not overlap.
Why is the second heart 12/51, not 13/52? Because the first heart left the deck. The second chance is conditional on that. Replacing the card would put 13/52 back.
Is 1/17 the chance of two hearts in a five-card hand? No. The 1/17 figure is the chance the first two cards are both hearts. A five-card hand is a different tree, with more cards and a combination count.