Multi-Event Probability Calculator | AND, OR, None, Exactly-One, Expected Count at Once
Enter the probabilities of several independent events and see the combined odds from five angles side by side. Comparing "everything succeeds" against "at least one succeeds" makes this useful for risk estimates on dependent tasks, win/loss scenarios, and reliability checks. Add or remove 2 to 10 events dynamically and label each one freely.
💡 About This Tool
Multiplying probabilities in your head is where intuition breaks down. Run five steps that each succeed 90% of the time and overall success drops to about 59%. Pull a lottery ticket with a 5% win rate thirty times and your chance of at least one win climbs to roughly 79%. AND (everything happens) and OR (at least one happens) use completely different formulas, and mixing them up throws an estimate off by orders of magnitude.
Assuming independent events, this tool computes five metrics together. All-occur is the product of the probabilities (p1 x p2 x ... x pn). At-least-one comes from the complement: 1 - (1-p1)(1-p2)...(1-pn). None-occur is that complement itself, exactly-one sums every case where a single event fires while the rest stay silent, and expected count is the plain sum of the probabilities (sum of pi). It fits any scenario you can treat as independent, including dependent project tasks, campaign hit-or-miss odds, and concurrent hardware failures.
The math assumes all events are mutually independent. When events are correlated (one happening makes another more likely), the figures will diverge from reality, so treat the outputs as a guide rather than a guarantee.
🧐 Frequently Asked Questions
Q. What is the difference between AND and OR? A. AND is the chance everything happens at once (a product, so it shrinks). OR is the chance at least one happens (closer to a sum, so it grows). For the same set of events, the two values differ sharply.
Q. What happens if I chain 90% five times? A. All succeed (AND) is about 59%, and the chance of at least one failure is about 41%. The more steps you add, the faster overall success collapses.
Q. What does "exactly one" measure? A. The probability that precisely one event fires while every other stays silent. It excludes both the all-fire and the all-quiet cases, isolating the single-hit scenario.
Q. What does an expected count of 1.00 mean? A. The average number of events that occur if you repeat the trial many times. It is the simple sum of the probabilities and acts as the central prediction for how many will fire.
Q. How many events can I add? A. Two to ten. With fewer than one event the combined odds are undefined, so outputs stay blank. At ten the add button is disabled.
Q. Can I use it for correlated events? A. The calculation assumes independence, so strongly correlated events introduce error. To model correlation you would need conditional probability or covariance handled separately.
📚 Fun Facts
The gap between intuition and probability has a name: the conjunction fallacy, made famous by Tversky and Kahneman's "Linda problem." People rate the probability of two conditions together as higher than a single condition, which directly contradicts the fact that an AND probability can never exceed any of its parts.
The "how many tries until I win" question is pure OR math. Even at a 1% win rate, a hundred attempts only give about a 63% chance of at least one win (1 - 0.99^100), so "a hundred tries is a sure thing" is a myth. Label events "try 1," "try 2," and so on, and the relationship between attempts and odds becomes concrete on screen.