Conditioning on the First Event
Split on what happens first, then notice the game often restarts.
What is it?
Condition on the outcome of the first round; if the game returns to its starting position, you get an equation in the unknown probability.
Why is it useful?
Alternating-turn games and "who wins first" questions are solved in two lines this way.
When should you think of it?
Use it when players take turns or a round can end or reset the game.
Think of this method when…
- Players alternate turns until someone succeeds
- A round either ends the game or restarts it
- "Probability that A wins"
- Infinite sums you would rather not write
Core idea
, solved for .
Step-by-step playbook
- 1List what can happen in the first round.
- 2Identify which outcomes end the game and which reset it.
- 3Write the equation for the target probability.
- 4Solve.
Worked example
Worked example
A and B take turns shooting at a target, A first, each hitting with probability . The first hit wins. What is ?
- 1
A wins immediately with . Both miss with , which restarts the game.
- 2
So .
- Answer:
Common mistakes
- Forgetting that the reset branch requires both players to miss.
- Summing a geometric series with the wrong ratio.
Would you use this method?
You roll a die once. What is the probability of a 6?
Practice this method
2 QuantSolve questions are tagged with this method.
Practice this method