The Core Probability Formula
For situations where every outcome is equally likely, the formula is straightforward:
Rolling a 4 on a fair die:
P(4) = 1/6 ≈ 0.167 ≈ 16.7%
You can express a probability as a fraction, a decimal, or a percentage - they're just three notations for the same number.
Reading Probability Values
| Value | Meaning | Example |
|---|---|---|
| 0 | Impossible | Rolling a 7 on a standard die |
| 0 to 0.5 | Unlikely | Flipping tails twice in a row (0.25) |
| 0.5 | Even chance | Flipping heads once |
| 0.5 to 1 | Likely | Rolling below 5 on a die (0.67) |
| 1 | Certain | Rolling below 7 on a standard die |
Complementary Events
An event's complement is everything else that could happen instead of it, and the two probabilities always add up to exactly 1:
P(rolling a 6) = 1/6
P(not rolling a 6) = 1 − 1/6 = 5/6
This shortcut solves a lot of problems faster than tackling them head-on - work out the probability of what you don't want, then subtract that from 1.
Independent Events - The Multiplication Rule
Events are independent when one has zero effect on the other, like two separate coin tosses. To find the probability of both happening, multiply:
P(heads then tails) = 1/2 × 1/2 = 1/4
Mutually Exclusive Events - The Addition Rule
Mutually exclusive events can't happen at the same time - you can't roll a 2 and a 5 on one single die roll. For these, add the probabilities together:
P(rolling a 2 or 5) = 1/6 + 1/6 = 2/6 = 1/3
Events That Can Overlap
When two events are able to happen together, you need to subtract that overlap so it isn't counted twice:
Drawing a heart or a queen from a 52-card deck:
P(heart) = 13/52, P(queen) = 4/52, P(queen of hearts) = 1/52
P(heart or queen) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13
Dependent Events and Conditional Probability
Events become dependent when the first one changes the odds for the second - drawing cards without putting them back is the standard example:
Drawing 2 aces from a deck without replacement:
P(1st ace) = 4/52
P(2nd ace | 1st was ace) = 3/51
P(both aces) = 4/52 × 3/51 = 12/2652 ≈ 0.0045
Theoretical vs. Experimental Probability
| Type | How it works | Example |
|---|---|---|
| Theoretical | Calculated from known outcomes | P(heads) = 1/2 by logic |
| Experimental | Measured by running trials | Flip a coin 100 times, count heads |
Run enough trials and experimental results start converging on the theoretical value - that's the Law of Large Numbers at work, not coincidence.
Quick Recap
The rules boil down to a short list: P(event) = favourable outcomes over total outcomes. For a complement, P(not A) = 1 − P(A). Independent events multiply together. Mutually exclusive events add together. Events that can overlap get added, then have that overlap subtracted back out. Dependent events multiply using conditional probability. Between these, you can solve the large majority of probability problems that show up in school, and if you'd rather skip the arithmetic, SolverCalc's calculators can check your work.