Probability Calculator

How to Calculate Probability — Basic Formula

Probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes: $P = \frac{\text{favorable}}{\text{total}}$. It's always a number between 0 (impossible) and 1 (certain), often expressed as a percentage. Rolling a 4 on a six-sided die has a probability of 1/6 ≈ 16.7%.

Permutations vs Combinations — When to Use Each

Use a permutation when the order of selection matters — like ranking runners 1st, 2nd, and 3rd. Use a combination when order doesn't matter — like picking 3 toppings for a pizza.

$${}_nP_r = \frac{n!}{(n-r)!} \qquad {}_nC_r = \frac{n!}{r!(n-r)!}$$

Because permutations count every possible ordering as distinct, ${}_nP_r$ is always at least as large as ${}_nC_r$ for the same $n$ and $r$.

Worked Example: Choosing Pizza Toppings

With 8 toppings available, how many ways can you choose 3, if order doesn't matter? That's a combination: ${}_8C_3 = \frac{8!}{3!(8-3)!} = 56$. If you were instead ranking your top 3 favorite toppings out of 8 in order of preference (1st, 2nd, 3rd), order matters, so it's a permutation: ${}_8P_3 = \frac{8!}{(8-3)!} = 336$ — six times larger, since each group of 3 toppings can be arranged in $3! = 6$ different orders.

Independent vs Mutually Exclusive Events

Independent events don't influence each other — the outcome of a coin flip doesn't change the odds of a dice roll. For independent events, $P(A \text{ and } B) = P(A) \times P(B)$. Mutually exclusive events can't happen at the same time — rolling a 2 and rolling a 5 on the same die roll are mutually exclusive. For mutually exclusive events, $P(A \text{ or } B) = P(A) + P(B)$. These are opposite concepts with correspondingly different formulas.

How to Calculate Odds

Odds compare favorable outcomes directly to unfavorable ones, rather than to the total: odds for = favorable : unfavorable, odds against = unfavorable : favorable. A probability of 3/10 (3 favorable out of 10 total) becomes odds of 3:7 in favor, since there are 7 unfavorable outcomes. This is also why odds and probability are easy to confuse but aren't interchangeable — "3 to 7 odds" and "30% probability" describe the exact same situation using two different reference points, and mixing them up when reading betting lines or risk statistics is a common source of confusion.

Expected Value — What "Average Outcome" Really Means

Expected value multiplies each possible outcome by its probability and sums the results, giving the long-run average you'd see if you repeated an event many times. A fair coin flip paying 10 dollars for heads and nothing for tails has an expected value of $0.5 \times 10 + 0.5 \times 0 = 5$ dollars per flip — not a value you'd ever actually see on a single flip, but the average if you flipped it thousands of times. Expected value is the foundation of how casinos, insurers, and investors price risk.

20 Common Probability Problems, Worked Step by Step

Five worked examples for each problem type below — same structure as the calculator above, so you can match your own problem to the closest one.

Single Event — 5 Worked Examples

1. Coin flip. One flip of a fair coin — what's the probability it lands Heads? $P = \frac{1}{2} = 50\%$.
2. Rolling an even number. One roll of a fair six-sided die — probability of rolling an even number? Favorable outcomes: 2, 4, 6. $P = \frac{3}{6} = \frac{1}{2} = 50\%$.
3. Drawing a heart. One card drawn from a standard 52-card deck — probability it's a heart? $P = \frac{13}{52} = \frac{1}{4} = 25\%$.
4. Raffle ticket. You hold 1 ticket in a raffle where 250 tickets were sold — probability yours is drawn? $P = \frac{1}{250} = 0.4\%$.
5. Sum of 7 on two dice. Roll two six-sided dice together as a single event — probability the faces add up to 7? The sample space has $6 \times 6 = 36$ equally likely outcomes; the pairs $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$ sum to 7. $P = \frac{6}{36} = \frac{1}{6} \approx 16.7\%$ — the most likely sum on two dice, since more pairs produce it than any other total.

Compound Events — 5 Worked Examples

1. AND, independent. Draw one card from a full deck and, separately, flip a fair coin — probability the card is a King and the coin lands Heads? The events don't influence each other, so multiply: $P(\text{King}) \times P(\text{Heads}) = \frac{4}{52} \times \frac{1}{2} = \frac{1}{26} \approx 3.85\%$.
2. OR, mutually exclusive. Rolling one die — probability of getting a 2 or a 5? A single roll can't land on both, so add: $P(2) + P(5) = \frac{1}{6} + \frac{1}{6} = \frac{1}{3} \approx 33.3\%$.
3. AND, independent, two dice. Roll two dice — probability both show a 6? $P = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \approx 2.78\%$.
4. Complement. A forecaster gives a 70% chance of rain ($P(A) = 0.7$) — probability it does not rain? $P(\text{not } A) = 1 - 0.7 = 0.3 = 30\%$.
5. OR, mutually exclusive, cards. One card drawn from a deck — probability it's a King or a Queen? Neither can happen at once, so add: $P = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13} \approx 15.4\%$.

The giveaway for which rule to use: "and" with independent events multiplies; "or" with mutually exclusive events adds. Mixing the two up — adding when you should multiply, or vice versa — is the single most common mistake in compound probability.

Permutations — 5 Worked Examples

1. Electing club officers. A club of 12 members elects a president, vice-president, and treasurer — one person per role. Order matters (president ≠ treasurer), so this is a permutation of 12 items taken 3 at a time: ${}_{12}P_3 = 12 \times 11 \times 10 = 1{,}320$ possible slates.
2. Race medals. 8 runners finish a race — in how many orders can gold, silver, and bronze be awarded? ${}_{8}P_3 = 8 \times 7 \times 6 = 336$.
3. 4-digit PIN, no repeats. How many 4-digit PINs can be formed from the digits 0–9 with no digit repeated? ${}_{10}P_4 = 10 \times 9 \times 8 \times 7 = 5{,}040$.
4. Arranging books on a shelf. In how many ways can 5 different books be lined up on a shelf? Every book is used, so this is ${}_5P_5 = 5! = 120$.
5. Front-row photo lineup. A photographer arranges 4 of 9 people in a row, left to right — position matters. ${}_{9}P_4 = 9 \times 8 \times 7 \times 6 = 3{,}024$.

Combinations — 5 Worked Examples

1. Forming a committee. From 15 volunteers, a 4-person committee is chosen — every member has the same role, so order doesn't matter: ${}_{15}C_4 = \frac{15!}{4! \times 11!} = 1{,}365$ possible committees.
2. Lottery numbers. A lottery draws 6 numbers from 49, order irrelevant — how many possible tickets? ${}_{49}C_6 = 13{,}983{,}816$ — which is why lottery jackpots stay unclaimed for so long.
3. 5-card poker hand. How many different 5-card hands can be dealt from a standard 52-card deck? ${}_{52}C_5 = 2{,}598{,}960$.
4. Ice cream scoops. Choosing 2 scoops of ice cream from 10 available flavors, no repeats, order doesn't matter (a cone with vanilla-then-chocolate is the same as chocolate-then-vanilla): ${}_{10}C_2 = 45$.
5. Study group. Forming a 3-student study group from a class of 20: ${}_{20}C_3 = 1{,}140$.

Combinations always shrink the count compared to the equivalent permutation, by a factor of $r!$ — the committee example above has ${}_{15}P_4 = 32{,}760$ orderings, but only $1{,}365 = 32{,}760 / 4!$ distinct groups, since each group of 4 people can be arranged in 24 different orders that a combination treats as identical.

A Brief History of Probability Theory

Probability as a mathematical discipline is often traced to a 1654 exchange of letters between Blaise Pascal and Pierre de Fermat, prompted by a gambler's question about how to fairly split the stakes of an interrupted game of chance — a problem now known as the "problem of points." Their correspondence laid the groundwork for treating chance as something that could be reasoned about systematically rather than left to superstition. Christiaan Huygens formalized much of this work in a 1657 treatise, widely considered the first published textbook on probability. The field matured significantly in the following centuries: Jacob Bernoulli's posthumously published "Ars Conjectandi" (1713) introduced the law of large numbers, and Pierre-Simon Laplace's work in the early 1800s established much of the classical framework — including the basic favorable-over-total formula — still taught today. Andrey Kolmogorov's 1933 axiomatization finally placed probability theory on rigorous mathematical footing, defining it in terms of set theory and measure theory rather than intuition alone.

Probability Terms You Should Know

Sample Space — the complete set of all possible outcomes of an experiment, such as the 6 faces of a die or the 52 cards in a deck.

Event — any subset of the sample space you're measuring the probability of, from a single outcome to a broad category of outcomes.

Complement — everything in the sample space that is not the event in question; its probability is always 1 minus the event's probability.

Conditional Probability — the probability of an event given that another event has already occurred, written $P(A \mid B)$, which can differ significantly from the unconditional probability of $A$ alone.

Random Variable — a variable whose value is the numerical outcome of a random event, forming the basis for calculating expected value and variance.