Probability

Finding the Chance of an Event, Alone or Combined

Probability measures how likely an event is, as a number between 0 (impossible) and 1 (certain). Enter a count of favorable and total outcomes to find a single event’s probability and its complement, or enter two independent events’ probabilities to find the chance both happen, either happens, or neither happens.

The Formula

Single event:

$$\vC{\text{Probability}} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$$

The complement (the chance the event does NOT happen) is always $1 - \vC{\text{Probability}}$.

Two independent events (where one event happening doesn’t change the chance of the other):

• $P(\text{both occur}) = P(\vA{A}) \times P(\vB{B})$
• $P(\text{either occurs}) = P(\vA{A}) + P(\vB{B}) - P(\vA{A}) \times P(\vB{B})$ — adding the two directly would double-count the overlap where both happen, so it’s subtracted back out once (the inclusion-exclusion principle).
• $P(\text{neither occurs}) = (1 - P(\vA{A})) \times (1 - P(\vB{B}))$

Worked Example

Single event: rolling an even number on a standard 6-sided die.

1. Favorable outcomes: 3 (2, 4, 6) out of 6 total: $3 \div 6 = \vC{50\%}$.
2. Complement: $1 - \vC{50\%} = 50\%$ (rolling an odd number).

Two independent events: flipping a coin and getting heads (P = 50%) AND rolling a die and getting a 6 (P ≈ 16.67%):

1. P(both) = 0.5 × 0.1667 ≈ 8.33%
2. P(either) = 0.5 + 0.1667 − 0.0833 ≈ 58.33%
3. P(neither) = 0.5 × 0.8333 ≈ 41.67%

Key Factors to Consider

• The independent-events formulas here only apply when one event’s outcome genuinely doesn’t affect the other’s. A coin flip and a die roll are classic independent events — but drawing two cards from a deck WITHOUT replacement is NOT independent, since removing the first card changes the odds for the second. Applying these formulas to dependent events gives an incorrect result.
• A probability of 0 means impossible and 1 means certain — every real probability falls somewhere between those two extremes. This 0-to-1 scale (or equivalently 0% to 100%) is the universal convention across probability and statistics, which is why this calculator’s results always fall within that range.
• The “gambler’s fallacy” is a common misconception this calculator’s math directly contradicts. Believing that a fair coin is “due” for tails after several heads in a row misunderstands independence — each flip has the exact same 50% probability regardless of what happened before, since previous flips have no actual effect on a fair coin’s next result.
• Classical probability (favorable outcomes divided by total outcomes) assumes every outcome is equally likely. This assumption holds for a fair die or a fair coin, but not for every real-world scenario — a weighted die or a biased coin would need actual observed frequencies instead of this simple counting method to estimate its true probabilities.

Common Mistakes

• Using the independent-events formulas on events that aren’t actually independent. Drawing two cards from a deck without replacing the first one is a classic example — removing the first card changes the odds for the second, so treating them as independent overstates or understates the real combined probability.
• Simply adding P(A) and P(B) to find the chance either occurs. That double-counts the case where both happen — the overlap has to be subtracted back out once, which is exactly what the inclusion-exclusion formula above does.
• Believing a fair coin or die is “due” for a particular result after a streak. Each independent flip or roll has the exact same probability every time, regardless of what happened on previous flips — this misconception (the “gambler’s fallacy”) is directly contradicted by the independence assumption these formulas rely on.

Useful to Know

• Working with a full dataset of numbers rather than a single probability? Statistics Calculator calculates mean, median, mode, and standard deviation.
• Need to count arrangements or selections instead of a probability? Permutation and Combination Calculator calculates permutations and combinations.
• Want to try rolling actual dice rather than just calculating the odds? Dice Roller Calculator rolls any number of dice with any number of sides.