Probability Calculator
Calculate combinations, permutations, Bayes' Theorem, binomial probability, and more with step-by-step solutions and interactive visualizations. Perfect for GCSE and A-Level students.
Combinations (nCr)
Order doesn't matter
nCr counts selections where order doesn't matter (e.g., choosing a committee)
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What is Probability?
Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). It is fundamental to statistics, data science, and everyday decision-making.
Basic Probability Formula
Example: P(rolling a 6) = 1/6 ≈ 0.167 = 16.7%
The probability scale runs from 0 (impossible) to 1 (certain). A fair coin has P(Heads) = 0.5, meaning it will land heads about half the time over many flips.
The Laws of Probability
Four essential rules that govern all probability calculations:
Addition Rule (OR)
The probability of A or B occurring. We subtract P(A∩B) to avoid double-counting outcomes in both events.
For mutually exclusive events (cannot happen together): P(A∪B) = P(A) + P(B)
P(Heart or King) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13
Multiplication Rule (AND)
The probability of both A and B occurring. Multiply P(A) by the conditional probability of B given A.
For independent events: P(A∩B) = P(A) × P(B)
P(two Aces in a row without replacement) = 4/52 × 3/51 = 12/2652 = 1/221
Complement Rule (NOT)
The probability of A NOT happening. Since all probabilities sum to 1, the complement is simply 1 minus the probability.
Essential shortcut: P(at least one) = 1 - P(none)
If P(rain) = 0.7, then P(no rain) = 1 - 0.7 = 0.3
Bayes' Theorem (UPDATE)
Updates a prior belief with new evidence to get a posterior probability. Converts P(B|A) into P(A|B) — crucial when you know the 'reverse' conditional.
A-Level topic: used in medical testing, spam filtering, and machine learning
P(Disease|Positive) = 0.95 × 0.01 / 0.05 = 0.19 (only 19% chance despite positive test)
Permutations vs Combinations
The most common source of confusion in combinatorics. Use this table to decide which to use:
| Property | Permutations (nPr) | Combinations (nCr) |
|---|---|---|
| Order | Matters | Does NOT matter |
| Formula | n! / (n-r)! | n! / (r!(n-r)!) |
| Example (3 from 5) | 60 | 10 |
| Real-world | Race placements, passwords | Lottery numbers, committees |
| Relationship | nPr = nCr × r! | nCr = nPr / r! |
| Always | nPr ≥ nCr | nCr ≤ nPr |
Quick Reference: Factorial Values
0!
1
1!
1
2!
2
3!
6
4!
24
5!
120
6!
720
7!
5,040
8!
40,320
9!
362,880
10!
3,628,800
12!
479,001,600
Types of Probability
There are different ways to determine or interpret probability:
Theoretical Probability
Based on equally likely outcomes. Calculated by reasoning about the situation without conducting experiments.
P(heads on fair coin) = 1/2 = 0.5
Experimental Probability
Based on observed frequency from actual experiments or data collection (relative frequency).
Flipped coin 100 times, got 47 heads: P ≈ 0.47
Subjective Probability
Based on personal judgement, experience, or belief. Used when no data or equally likely model exists.
"I think there's a 60% chance the team will win"
Conditional Probability
The probability of an event given that another event has already occurred. Changes the sample space.
P(King | Face card drawn) = 4/12 = 1/3
Real-World Applications
Probability is used everywhere — here are some key areas:
Medicine
Bayes' Theorem calculates the true probability of disease given a positive test result. A 95% accurate test on a rare disease (1% prevalence) yields only ~19% true positive rate.
Bayes' Theorem in diagnosticsInsurance
Actuaries use probability distributions to estimate risk and set premiums. They calculate the expected value of claims to ensure profitability while remaining competitive.
Expected value for premiumsGames & Gambling
Combinatorics determines lottery odds (e.g., 6 from 49 = 13,983,816 combinations). Card game probabilities use conditional probability and Bayes for strategy.
Combinations for oddsWeather Forecasting
"70% chance of rain" means that in similar atmospheric conditions, rain occurred 70% of the time historically. This is experimental/frequentist probability.
Experimental probabilityCommon Mistakes in Probability
Avoid these frequent errors when solving probability questions in GCSE and A-Level exams:
Adding probabilities for non-mutually exclusive events
P(A∪B) = P(A) + P(B) is ONLY correct when A and B cannot happen together. Otherwise you must subtract P(A∩B) to avoid double-counting.
Always use P(A∪B) = P(A) + P(B) - P(A∩B). Only drop the last term if events are mutually exclusive.
Confusing nPr and nCr (wrong formula)
Using permutations when you need combinations (or vice versa) gives answers that are off by a factor of r!. This is the single most common combinatorics error.
Ask: "Does the order matter?" Arrangements → nPr. Selections → nCr. When unsure, nCr is more common in exam questions.
Assuming P(A|B) = P(B|A) (Bayes' confusion)
P(Disease|Positive) is NOT the same as P(Positive|Disease). A test that is 95% accurate at detecting disease does NOT mean a positive result gives 95% chance of having it.
Use Bayes' Theorem: P(A|B) = P(B|A) × P(A) / P(B). The prior P(A) matters enormously.
Getting a probability > 1 or < 0
All probabilities must be between 0 and 1 inclusive. This is a useful sanity check — if your answer is 1.3 or -0.2, you made an arithmetic error.
Always verify: 0 ≤ P(E) ≤ 1. Also check: P(A) + P(A') = 1 and P(A∪B) ≤ P(A) + P(B).
Treating dependent events as independent
Drawing cards without replacement makes events dependent. P(2nd Ace) changes depending on whether the 1st card was an Ace (3/51 vs 4/51).
Check: are the events truly independent? If one outcome affects the next, use P(A∩B) = P(A) × P(B|A), not P(A) × P(B).
Confusing "at least one" with "exactly one"
"At least one" means 1, 2, 3, or more. Calculating each case and adding is tedious and error-prone.
Use the complement trick: P(at least one) = 1 - P(none). Much simpler and less prone to mistakes.
Worked Examples
Practice with these GCSE and A-Level style probability problems:
Example 1: Probability with Dice
A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?
Solution:
Favorable outcomes: 5, 6 → 2 outcomes
Total outcomes: 1, 2, 3, 4, 5, 6 → 6 outcomes
P(greater than 4) = 2/6 = 1/3
P = 1/3 ≈ 0.333 = 33.3%
Example 2: Choosing a Team
A teacher needs to choose 3 students from a class of 10 to represent the school. How many different teams are possible?
Solution:
Order does NOT matter (any 3 students form the same team)
Use combinations: 10C3 = 10! / (3! × 7!)
= 3,628,800 / (6 × 5,040)
= 3,628,800 / 30,240
10C3 = 120 different teams
Example 3: Binomial Distribution
A fair coin is flipped 7 times. What is the probability of getting exactly 4 heads?
Solution:
n = 7, k = 4, p = 0.5
P(X=4) = 7C4 × 0.5⁴ × 0.5³
7C4 = 7! / (4! × 3!) = 35
P(X=4) = 35 × 0.0625 × 0.125
P(X=4) = 0.2734 = 27.34%
Example 4: Medical Test Accuracy
A disease affects 1% of the population. A test is 95% accurate (positive when disease present) and has a 4% false positive rate. If a person tests positive, what is the probability they have the disease?
Solution:
P(D) = 0.01, P(+|D) = 0.95, P(+|D') = 0.04
P(+) = P(+|D)×P(D) + P(+|D')×P(D')
P(+) = 0.95×0.01 + 0.04×0.99 = 0.0095 + 0.0396 = 0.0491
P(D|+) = P(+|D)×P(D) / P(+) = 0.0095 / 0.0491
P(D|+) ≈ 0.1935 = 19.35%
Despite a 95% accurate test, a positive result only means a 19.35% chance of disease. This counterintuitive result is why Bayes' Theorem matters.
Exam Tips
Order Matters?
Ask yourself: does the order of selection matter? If yes, use nPr. If no, use nCr.
Remember 0! = 1
By definition, 0! = 1. This is essential for formulas like nC0 = 1 and nCn = 1.
Subtract for Union
P(A∪B) = P(A) + P(B) - P(A∩B). Subtract to avoid double-counting the overlap.
Multiply for Independent
For independent events: P(A∩B) = P(A) × P(B). Don't add them!
Check Probability Range
All probabilities must be between 0 and 1. If you get a value outside this range, recheck your work.
Complement Trick
P(at least one) = 1 - P(none). Often much easier than calculating every case directly.
Frequently Asked Questions
What is the difference between permutations and combinations?
Permutations (nPr) count arrangements where ORDER MATTERS — like choosing 1st, 2nd, and 3rd place winners. Combinations (nCr) count selections where ORDER DOES NOT MATTER — like choosing a committee. nPr = n!/(n-r)! and nCr = n!/(r!(n-r)!). nPr is always ≥ nCr.
How do I know when to use nPr vs nCr?
Ask: "Does the order matter?" If you're arranging things in specific positions (race placements, seating, passwords), use nPr. If you're just selecting a group (committee, lottery, hand of cards), use nCr.
What is conditional probability P(A|B)?
P(A|B) is the probability of A given B has occurred. Formula: P(A|B) = P(A∩B) / P(B). Example: P(King | Face card) = (4/52) / (12/52) = 4/12 = 1/3.
When do I use the binomial probability formula?
Use it when you have: (1) a fixed number of independent trials, (2) two outcomes per trial (success/failure), (3) constant probability per trial, and (4) you want P(exactly k successes). Examples: coin flips, MCQ guessing, quality control.
Why is 0! equal to 1?
0! = 1 by definition. It ensures formulas work: nCn = n!/(n!×0!) = 1 and nC0 = n!/(0!×n!) = 1. There is exactly one way to choose all items, and one way to choose nothing.
What is the addition rule for probability?
P(A∪B) = P(A) + P(B) - P(A∩B). We subtract P(A∩B) because outcomes in both events are counted twice. For mutually exclusive events: P(A∪B) = P(A) + P(B).
What is Bayes' Theorem and when do I use it?
Bayes' Theorem updates probability with new evidence: P(A|B) = P(B|A) × P(A) / P(B). Use it when you know P(B|A) and want P(A|B). Classic example: finding P(Disease | Positive test) when you know the test's sensitivity and the disease prevalence.
What is the difference between mutually exclusive and independent events?
Mutually exclusive: cannot happen together (P(A∩B) = 0). Example: rolling 3 AND 5 on one die. Independent: one doesn't affect the other (P(A∩B) = P(A)×P(B)). Example: two separate coin flips. Important: mutually exclusive events are NEVER independent (unless one has P = 0).
How do I calculate expected value?
E[X] = Σ xᵢ × P(xᵢ). Multiply each outcome by its probability and sum. For a fair die: E[X] = 1(1/6) + 2(1/6) + ... + 6(1/6) = 3.5. It represents the long-run average outcome.
What does "at least one" mean in probability?
"At least one" = 1 or more. The easiest method: P(at least one) = 1 - P(none). Example: P(at least one 6 in 3 rolls) = 1 - (5/6)³ = 1 - 125/216 ≈ 0.421.
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