Finding an unknown branch probability algebraically

A recurring Excellence step: a tree has an unknown branch probability (x, or 2x), solvable ONLY by setting up an equation from a given OVERALL probability.

The method

Assign the unknown branch a variable (commonly x), and any related branch in terms of it (e.g. 2x if one is stated to be twice the other).
Write an expression for the OVERALL probability the question already tells you, by multiplying along each relevant path and adding across paths — using x in place of the unknown branch(es).
Set this expression EQUAL to the given overall probability, forming a solvable equation in x.
Solve for x, then substitute back to find every other unknown branch probability the question actually asked for.

Worked example Finding an unknown win probability from an overall rate A game is played with friends 70% of the time (win rate 10%) and alone 30% of the time (win rate x). The overall win rate is 17.5%. Find x. SET UP THE EQUATION: overall win rate = P(friends)×P(win|friends) + P(alone)×P(win|alone) = 0.70 × 0.10 + 0.30 × x. SUBSTITUTE the given overall rate: 0.175 = 0.07 + 0.30x. SOLVE: 0.30x = 0.105 → x = 0.35. CHECK: 0.70 × 0.10 + 0.30 × 0.35 = 0.07 + 0.105 = 0.175 ✓ — matches the given overall rate. ✓ Always CHECK the solved value against the given overall probability before moving on — a sign error in setting up the equation is easy to make and easy to catch this way. ✕

Solving for x without using the OVERALL probability given

This type of question cannot be solved from the tree structure alone — the unknown branch is only findable because the question GIVES an overall probability that constrains it. Skipping straight to "the branches must sum to 1" on the wrong level of the tree, instead of using the stated overall probability, leads to an unsolvable or wrong equation.

Step	What to do
1. Assign the unknown	Let the unknown branch = x (and any related branch, e.g. 2x)
2. Express the overall probability	Multiply along each path, add across paths, using x
3. Set equal to the given value	Form a solvable equation
4. Solve, then check	Substitute back to confirm it matches the given overall probability

Practice for this standard covers the whole topic — flashcards and quiz.