What Is the Probability Formula?
By the Calculator.hair team · Updated · 8 min read
Short answer
The basic probability formula is P(A) = favorable outcomes ÷ total possible outcomes, when all outcomes are equally likely. The chance of rolling a six on a fair die is 1 ÷ 6 ≈ 0.1667, and the chance of drawing an ace from a 52-card deck is 4 ÷ 52 ≈ 0.0769. A probability always falls between 0 (impossible) and 1 (certain).
Key takeaways
- Basic probability is favorable outcomes divided by total outcomes, as long as the outcomes are equally likely.
- Multiply probabilities for AND, but only use P(A) × P(B) when the events are independent.
- For OR, add the probabilities and subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B).
- Conditional probability is P(A | B) = P(A and B) ÷ P(B), and P(A | B) usually differs from P(B | A).
- Permutations count ordered arrangements and combinations count unordered groups, so nPr is always at least as large as nCr.
What is the formula for probability?
The formula for probability is P(A) = number of favorable outcomes ÷ total number of possible outcomes. P(A) is read as "the probability of event A," and an event is any outcome or group of outcomes you care about. This counting formula assumes every outcome is equally likely, as with a fair die or a shuffled deck.
A result can be written as a fraction, decimal or percent: 1/4, 0.25 and 25% are the same value. The table shows the main rules used in this guide. Our probability calculator computes several of them for two independent events.
| Rule | Formula | Example | Result |
|---|---|---|---|
| Basic | P(A) = favorable ÷ total | Roll a six | 1/6 ≈ 0.1667 |
| Complement | P(not A) = 1 − P(A) | Don't roll a six | 5/6 ≈ 0.8333 |
| AND, independent | P(A and B) = P(A) × P(B) | Heads and a six | 1/12 ≈ 0.0833 |
| AND, dependent | P(A and B) = P(A) × P(B | A) | Two aces, no replacement | 1/221 ≈ 0.0045 |
| OR | P(A or B) = P(A) + P(B) − P(A and B) | Heart or ace | 16/52 ≈ 0.3077 |
| Conditional | P(A | B) = P(A and B) ÷ P(B) | Even, given over 3 | 2/3 ≈ 0.6667 |
| Binomial | P(k) = nCk × pᵏ × (1 − p)ⁿ⁻ᵏ | 3 heads in 5 flips | 10/32 = 0.3125 |
How do you calculate the probability of something not happening?
Subtract the probability of the event from 1: P(not A) = 1 − P(A). This is the complement rule, where the complement of an event means every outcome that is not in the event. The chance of not rolling a six is 1 − 1/6 = 5/6 ≈ 0.8333.
The rule makes "at least one" questions easier. To find the chance of at least one six in four rolls, find the chance of no sixes and subtract it from 1. P(no sixes) = (5/6)⁴ ≈ 0.4823, so P(at least one six) ≈ 1 − 0.4823 = 0.5177. The calculator's series tool returns about 0.5177 with P(A) = 0.1666666667 and 4 repeats. Simply adding 1/6 four times gives 0.6667, which is wrong because it counts rolls with more than one six more than once.
What is the AND rule in probability?
For two independent events, multiply their probabilities: P(A and B) = P(A) × P(B). Independent means that one event does not change the chance of the other. The chance of flipping heads and rolling a six is 1/2 × 1/6 = 1/12 ≈ 0.0833, which our calculator confirms.
When the first event changes the odds for the second, the events are dependent, and you need P(A and B) = P(A) × P(B | A). Drawing two aces without putting the first card back is 4/52 × 3/51 = 12/2,652 = 1/221 ≈ 0.0045, because only 3 aces remain among 51 cards. If you replace the first card and reshuffle, the draws become independent and the answer is 4/52 × 4/52 = 1/169 ≈ 0.0059.
Our calculator's two-event forms assume independence, so use the dependent version of this rule by hand. Before relying on the calculator, ask whether knowing one result would change your estimate of the other.
What is the OR rule in probability?
Add the two probabilities and subtract the chance of both: P(A or B) = P(A) + P(B) − P(A and B). The subtraction stops outcomes in both events from being counted twice. In probability, "or" normally includes the case where both happen.
Take the chance of drawing a heart or an ace. There are 13 hearts and 4 aces, but the ace of hearts is in both groups. So P = 13/52 + 4/52 − 1/52 = 16/52 ≈ 0.3077. Our calculator gives about 0.3077 with P(A) = 0.25 and P(B) = 0.0769230769 (1/13), and it applies here since suit and rank are independent. It also reports about 0.2885 for exactly one of the two and 0.6923 for neither.
If two events can't happen together, they are mutually exclusive, and the overlap is 0. The chance of rolling a 1 or a 2 on one die is 1/6 + 1/6 = 1/3. This is hand arithmetic, because our two-event calculator assumes independence, and mutually exclusive events with nonzero chances are never independent: if one happens, the other's chance drops to 0.
What is the conditional probability formula?
The conditional probability formula is P(A | B) = P(A and B) ÷ P(B). The bar is read as "given," so P(A | B) is the probability of A once you know B happened. Knowing B shrinks the set of possible outcomes to just those in B.
Roll one die and suppose you learn the result is over 3, so it is 4, 5 or 6. Two of those three are even. By the formula, P(even | over 3) = (2/6) ÷ (3/6) = 2/3. Without that information, the chance of an even number is 1/2. Since the two figures differ, these events are dependent.
Our calculator does not have a conditional probability form, so these results are hand arithmetic. A card example shows that the order of the condition matters: P(ace | heart) = (1/52) ÷ (13/52) = 1/13, while P(heart | ace) = (1/52) ÷ (4/52) = 1/4.
What is the binomial probability formula?
The binomial probability formula gives the chance of exactly k successes in n independent trials: P(k) = nCk × pᵏ × (1 − p)ⁿ⁻ᵏ. Here p is the chance of success on each trial, and nCk counts the ways to choose which trials succeed. The trials must be fixed in number, have two outcomes each and share the same p.
Two worked examples, using nCk from our permutation and combination calculator and the rest by hand:
- Exactly 3 heads in 5 fair flips: 5C3 = 10, so P = 10 × (1/2)³ × (1/2)² = 10/32 = 0.3125.
- Exactly 2 sixes in 5 rolls: 5C2 = 10, so P = 10 × (1/6)² × (5/6)³ = 1,250/7,776 ≈ 0.1608.
What is the difference between permutations and combinations?
Permutations count arrangements where order matters, while combinations count groups where order does not matter. The formulas are nPr = n! ÷ (n − r)! and nCr = n! ÷ (r! × (n − r)!), where n! (n factorial) means n × (n − 1) × ... × 1.
Awarding gold, silver and bronze among 10 runners has 720 permutations, because finishing order matters. Choosing a 3-person committee from the same 10 people has only 120 combinations. Each group of 3 can be ordered in 3! = 6 ways, and 720 ÷ 6 = 120.
Counting feeds straight into the basic formula. A 5-card poker hand is a combination, with 2,598,960 possible hands. Four of those are royal flushes, one per suit, so P(royal flush) = 4 ÷ 2,598,960 = 1 in 649,740. Combinations also confirm the two-aces result from earlier: 4C2 ÷ 52C2 = 6 ÷ 1,326 = 1/221.
| Scenario | n, r | Order matters? | Count |
|---|---|---|---|
| Top 3 finishers of 10 | 10, 3 | Yes (nPr) | 720 |
| 3-person committee from 10 | 10, 3 | No (nCr) | 120 |
| 3-digit code, digits may repeat | 10, 3 | Yes, with repetition (nʳ) | 1,000 |
| Two-card starting hand | 52, 2 | No (nCr) | 1,326 |
| 5-card poker hand | 52, 5 | No (nCr) | 2,598,960 |
What are common probability mistakes?
The best-known mistake is the gambler's fallacy, the belief that a streak makes the opposite result more likely. After five heads in a row, the chance of heads on the next fair flip is still 1/2, because the coin has no memory. A streak of 10 heads does have a low chance in advance, 1/1,024 ≈ 0.0009765625, but so does any other specific sequence of 10 flips. Over many flips, the share of heads tends toward 50% because the early streak is diluted by later flips, not because the coin corrects for it.
Other common errors are:
- Swapping P(A | B) and P(B | A). The chance that a card is an ace given it is a heart (1/13) differs from the chance it is a heart given it is an ace (1/4).
- Multiplying dependent events as if they were independent.
- Treating mutually exclusive and independent as the same thing. They are nearly opposite ideas.
- Forgetting to subtract the overlap in the OR rule.
- Confusing odds with probability. Odds of 1 to 5 mean a probability of 1/6, not 1/5.
How does probability connect to the normal distribution?
For measurements such as heights or test scores, probability comes from a continuous curve rather than from counting outcomes. Our calculator's normal distribution form shows that about 0.6827 of a normal distribution falls within one standard deviation of the mean. See how to calculate standard deviation for how that spread is measured.
Calculators in this guide
Frequently asked questions
What is the equation for probability?
P(A) = favorable outcomes ÷ total possible outcomes, for equally likely outcomes. Drawing a heart from a full deck is 13 ÷ 52 = 0.25.
What is the probability of rolling a 7 with two dice?
It is 6/36 = 1/6 ≈ 0.1667. Six of the 36 equally likely pairs add up to 7: 1-6, 2-5, 3-4, 4-3, 5-2 and 6-1.
When do you multiply and when do you add probabilities?
Multiply for AND, when both events must happen. Add for OR, when either event will do, and subtract the overlap if both can happen together.
Can a probability be greater than 1?
No. All probabilities fall from 0 to 1, or 0% to 100%. A result above 1 usually means overlapping outcomes were counted twice or the wrong total was used.
How do you know if two events are independent?
Check whether P(A and B) equals P(A) × P(B). Heart and ace pass, since 1/4 × 1/13 = 1/52, but even and over 3 on one die fail, since 1/2 × 1/2 = 1/4 and the true value is 1/3.