Conditional chain probability calculator
Enter the probability of each step, conditional on the ones before it. Get the joint probability of the whole chain and the contract price it implies.
| Step | P(step | previous steps), % | Running joint |
|---|
Formula used: P(chain) = P(A) × P(B | A) × P(C | A,B) × … — —
Fair price in cents = P(chain) × 100. Break-even decimal odds = 1 ÷ P(chain).
Why chains feel more likely than they are
Each additional step can only subtract. A step at 95% still shaves 5% off everything upstream of it, and six such steps together clear only 74% of the time. When you price a multi-stage market — a candidate winning a nomination and then a general election, a drug clearing three trial phases, a deal signing and then closing — the intuition to distrust is the one that anchors on the single most likely leg.
Getting the conditional part right
The number in each row is not the standalone probability of that step. It is the probability given that everything before it already happened, which is usually higher than the standalone figure, because early success is evidence the chain is a good one. Entering standalone numbers for correlated steps systematically underprices the chain.
Common questions
How do I calculate the probability that every step in a chain happens?
Multiply the probability of each step, where each step's number is conditional on the earlier steps already having happened. For four steps at 90%, 80%, 70% and 60%, the joint probability is 0.9 × 0.8 × 0.7 × 0.6 = 30.2%. The chain is always less likely than its weakest link.
Why does a market for a multi-stage event trade so cheaply?
Because probabilities multiply rather than average. Five steps that each look like a comfortable 85% still only clear together 44% of the time, so a contract on the full sequence fairly prices near 44¢. People overprice these chains because they judge the sequence by its most likely step.
Do the steps have to be independent for this to work?
No — the multiplication rule holds for any chain as long as each probability you enter is conditional on the previous steps succeeding. If you enter unconditional standalone probabilities for steps that are correlated, the result will be wrong, usually too low. Ask yourself for each row: given everything before it worked, how likely is this?