Statistics
How To Calculate Probability
Probability is the number of ways an event can happen divided by the number of equally likely outcomes. Everything else, from complements to combined events, is built from that one fraction.
Quick Answer
P = favourable outcomes / total outcomes
- P
- Probability of the event, always between 0 and 1
- n_fav
- Number of outcomes counted as the event happening
- n_total
- Total number of equally likely outcomes
- Pc
- Complement, the probability the event does not happen
Three favourable outcomes out of ten give a probability of 0.3, which is 30 percent, and a complement of 0.7. The odds in favour are 0.4285714286 to 1. Combining two independent events with probabilities 0.3 and 0.25 gives 0.075 for both happening together and 0.475 for at least one.
What Is Probability?
Probability measures how likely an event is, on a scale from zero for impossible to one for certain. For an experiment with a finite set of equally likely outcomes, it is computed as the number of outcomes that count as the event divided by the total number of outcomes. Three favourable results out of ten give a probability of 0.3.
The phrase equally likely is doing real work in that definition. It is what makes counting a valid method, because each outcome carries the same weight. A biased coin still has two outcomes, but they are not equally likely, and the counting rule no longer applies without modification.
Multiplying by one hundred converts a probability into a percentage, which is usually the more readable form. A probability of 0.3 is thirty percent. The conversion is trivial but worth doing deliberately, because a figure like 0.3 read as a percentage gives thirty percent while 0.3 read as a proportion gives a very different mental picture.
The complement is the probability that the event does not happen, and it is one minus the probability that it does. A probability of 0.3 leaves a complement of 0.7. Complements are useful because it is often far easier to count the ways something fails than the ways it succeeds, and the two always sum to one.
Odds are not the same as probability, although the two are routinely confused. Probability divides favourable outcomes by all outcomes, while odds in favour divide favourable outcomes by unfavourable ones. Three favourable out of ten gives a probability of 0.3 but odds of 0.4285714286 to 1, because there are seven unfavourable outcomes rather than ten total.
Independent events are combined by multiplication. If two events do not influence each other, the probability that both happen is the product of their individual probabilities. A 0.3 chance multiplied by a 0.25 chance gives 0.075, which is much smaller than either factor and reflects how quickly joint probabilities fall.
The probability that at least one of two independent events happens is one minus the probability that neither happens. That is one minus 0.7 times 0.75, which is 0.475. Working through the complement is far shorter than adding the individual probabilities and then correcting for the overlap.
Adding probabilities directly only works for mutually exclusive events, which cannot both occur. Two outcomes of the same dice roll are mutually exclusive, so their probabilities add. Two independent events are not, which is why the addition has to be corrected by subtracting the overlap, and why the complement route is often cleaner.
The expected number of occurrences in a run of trials is the probability multiplied by the number of trials. A probability of 0.1666667 for rolling a seven means a hundred rolls should produce about 16.66667 sevens. Expectation is a long-run average, not a prediction for any particular run.
Probabilities of complementary events always sum to one, and a value outside the range zero to one signals an arithmetic error rather than an unusual situation. That range check is a quick way to catch a reversed division or a forgotten subtraction before the wrong number is used for anything important.
Formula
P = n_fav / n_total
The definition for equally likely outcomes. The result always lies between zero and one, and multiplying by one hundred gives a percentage.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| n_fav | Favourable outcomes | outcomes | How many outcomes count as the event happening. |
| n_total | Total outcomes | outcomes | Every equally likely outcome in the sample space. |
| P | Probability | dimensionless | Between zero and one inclusive. |
P(not A) = 1 - P(A)
The probability that the event does not happen. Useful when counting the ways something fails is easier than counting the ways it succeeds.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P | Probability of the event | dimensionless | Between zero and one. |
| Pc | Complement | dimensionless | One minus the probability of the event. |
P(A and B) = P(A) x P(B)
Multiplies the probabilities when the events do not influence each other. For at least one happening, take one minus the product of the complements.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P(A) | Probability of A | dimensionless | The first event's probability. |
| P(B) | Probability of B | dimensionless | The second event's probability. |
| both | Joint probability | dimensionless | The chance both events happen. |
How To Calculate Probability
- 1
List the total number of equally likely outcomes
Count everything that could happen, not only the outcomes you care about. A single dice roll has six outcomes, and a pair of dice has thirty six.
- 2
Count the outcomes that count as the event
Rolling a seven has six ways out of thirty six, because one and six, two and five, and three and four each work in either order. Counting carefully is the whole of the method.
- 3
Divide to get the probability
Six divided by thirty six is 0.1666667. The result must lie between zero and one; anything outside that range means the counts were mixed up.
- 4
Multiply by one hundred for a percentage
A probability of 0.3 is thirty percent. Percentages are easier to compare at a glance, so it is worth writing both forms when the number will be read by someone else.
- 5
Combine events with the right rule
Multiply for independent events happening together, and use one minus the product of the complements for at least one. Adding works only for mutually exclusive outcomes.
Examples
Example 1: Three favourable outcomes out of ten
- Favourable outcomes
- 3
- Total outcomes
- 10
| Step | Calculation | Result |
|---|---|---|
| Probability | 3 / 10 | 0.3 |
| As a percentage | 0.3 x 100 | 30 |
| Complement | 1 - 0.3 | 0.7 |
| Odds in favour | 3 / (10 - 3) | 0.4285714286 |
Result: The probability is 0.3, or 30 percent, the complement is 0.7, and the odds in favour are 0.4285714286 to 1.
Example 2: Two independent events combined
- First event
- 0.3
- Second event
- 0.25
| Step | Calculation | Result |
|---|---|---|
| Both happening together | 0.3 x 0.25 | 0.075 |
| Neither happening | 0.7 x 0.75 | 0.525 |
| At least one happening | 1 - 0.525 | 0.475 |
Result: The chance both events happen is 0.075, while the chance that at least one happens is 0.475.
Example 3: Expected sevens in a hundred dice rolls
- Favourable outcomes
- 6
- Total outcomes
- 36
- Rolls
- 100
| Step | Calculation | Result |
|---|---|---|
| Probability of a seven | 6 / 36 | 0.1666667 |
| Expected sevens in a hundred rolls | 0.1666667 x 100 | 16.66667 |
Result: Rolling a seven has a probability of 0.1666667, so 100 rolls should produce about 16.66667 sevens.
Calculator
Probability
0.3
- Probability as a percent
- 30
- Complement
- 0.7
- Odds in favour
- 0.4286
- Both independent events
- 0.075
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the Probability calculator page.
Common Mistakes
Confusing probability with odds
Probability divides favourable outcomes by all outcomes, while odds divide favourable by unfavourable. Three of ten is a probability of 0.3 but odds of 0.4285714286 to 1, and treating the two as the same gives a materially different number.
Adding probabilities of independent events
Two independent events do not have probabilities that simply add. The joint chance is the product, and for at least one the correct route is one minus the product of the complements. Adding double-counts the overlap.
Assuming outcomes are equally likely when they are not
The counting rule only applies when every outcome carries the same weight. A weighted die has six faces but not six equal probabilities, so the favourable count over total count no longer gives the right answer.
Forgetting to subtract for the overlap
When adding the probabilities of two non-exclusive events, the overlap has been counted twice and must be subtracted once. Omitting that correction inflates the result and can push it above one.
Reporting a probability as a percentage without converting
A probability of 0.3 is thirty percent, not 0.3 percent. The conversion multiplies by one hundred, and skipping it understates the figure by two orders of magnitude.
Treating expected value as a prediction
An expectation of 16.66667 sevens over a hundred rolls is a long-run average. Any particular run of a hundred rolls will produce a whole number, often some distance from the expectation.
FAQ
What is the difference between probability and odds?
Probability is favourable outcomes divided by all outcomes, so it never exceeds one. Odds are favourable outcomes divided by unfavourable ones, so they can be larger than one. A probability of 0.3 corresponds to odds of 0.4285714286 to 1.
Can a probability be greater than one?
No. Probability runs from zero for impossible to one for certain, and a value outside that range always signals an arithmetic mistake such as a reversed division or a missing subtraction.
How do I find the probability of at least one of two events?
Take one minus the probability that neither happens. For events with probabilities 0.3 and 0.25 the chance of neither is 0.525, so the chance of at least one is 0.475. This avoids the double-counting that comes from adding directly.
When can I add probabilities instead of multiplying?
Add for mutually exclusive events that cannot both occur, such as two different outcomes of a single dice roll. Multiply for independent events that do not influence each other. If the events can overlap, adding requires subtracting the overlap.
What does a probability of zero or one mean?
Zero means the event cannot happen, and one means it is certain. Both are legitimate values rather than errors, and they mark the ends of the scale against which every other probability is compared.
How many times should an event happen in a run of trials?
The expected count is the probability multiplied by the number of trials. A probability of 0.1666667 over a hundred rolls gives an expectation of 16.66667, which is an average rather than a guarantee for any single run.
References
- [1]Wikipedia, Probability — https://en.wikipedia.org/wiki/Probability
- [2]Wikipedia, Complement (probability) — https://en.wikipedia.org/wiki/Complement_(probability)
- [3]Wikipedia, Independence (probability theory) — https://en.wikipedia.org/wiki/Independence_(probability_theory)