Simple Probability Calculator

Calculate the probability of an event by entering the number of favorable outcomes and total possible outcomes.

How many outcomes result in success?

Total number of all possible outcomes

Combinations Calculator (nCr)

Calculate the number of ways to choose r items from n items when order doesn't matter.

Total number of items to choose from

Number of items to select

Permutations Calculator (nPr)

Calculate the number of ways to arrange r items from n items when order matters.

Total number of items to arrange from

Number of positions to fill

Probability Rules Calculator

Calculate combined probabilities using AND, OR, and NOT rules.

Enter as decimal (0 to 1) or percentage will be converted

Enter as decimal (0 to 1)

Binomial Probability Calculator

Calculate the probability of exactly k successes in n independent trials.

How many times is the experiment repeated?

How many successes do you want?

Probability of success on a single trial (0 to 1)

Understanding Probability in UK Education

๐Ÿ“š What is Probability?

Probability is the branch of mathematics that deals with measuring the likelihood of events occurring. In the UK curriculum, probability is introduced at Key Stage 2 and developed through GCSE and A-Level. It's essential for understanding statistics, data analysis, risk assessment, and decision-making in everyday life.

๐Ÿ“— GCSE Mathematics

  • Calculate simple probabilities
  • Use sample space diagrams
  • Construct and use tree diagrams
  • Calculate relative frequency
  • Understand mutually exclusive events
  • Apply the addition rule P(A or B)
  • Work with independent events
  • Apply the multiplication rule P(A and B)

๐Ÿ“˜ A-Level Mathematics

  • Combinations and permutations
  • Binomial distribution
  • Normal distribution
  • Conditional probability
  • Discrete random variables
  • Expected value and variance
  • Probability density functions
  • Hypothesis testing

๐Ÿ“™ A-Level Further Maths

  • Poisson distribution
  • Geometric distribution
  • Continuous distributions
  • Central Limit Theorem
  • Bayes' Theorem
  • Markov chains
  • Monte Carlo methods
  • Advanced hypothesis testing

Essential Probability Formulas

Simple Probability

P(A) = n(A) / n(S)

Where n(A) is the number of favorable outcomes and n(S) is the total number of outcomes in the sample space.

Complement Rule

P(not A) = 1 - P(A)

The probability of an event not occurring equals 1 minus the probability it does occur.

Addition Rule (OR)

P(A โˆช B) = P(A) + P(B) - P(A โˆฉ B)

For mutually exclusive events: P(A โˆช B) = P(A) + P(B)

Multiplication Rule (AND)

P(A โˆฉ B) = P(A) ร— P(B|A)

For independent events: P(A โˆฉ B) = P(A) ร— P(B)

Combinations

โฟCแตฃ = n! / (r!(n-r)!)

The number of ways to choose r items from n items when order doesn't matter.

Permutations

โฟPแตฃ = n! / (n-r)!

The number of ways to arrange r items from n items when order matters.

Conditional Probability

P(A|B) = P(A โˆฉ B) / P(B)

The probability of A occurring given that B has already occurred.

Binomial Probability

P(X=k) = โฟCโ‚– ร— pแต ร— (1-p)โฟโปแต

Probability of exactly k successes in n independent trials.

Common Probability Reference Values

Scenario Probability Percentage Odds
Coin flip - Heads 0.5 50% 1:1
Roll a 6 on standard die 0.1667 16.67% 1:5
Roll an even number 0.5 50% 1:1
Drawing an ace from a deck 0.0769 7.69% 4:48 (1:12)
Drawing a heart from a deck 0.25 25% 13:39 (1:3)
Drawing a face card 0.2308 23.08% 12:40 (3:10)
Two heads in two coin flips 0.25 25% 1:3
At least one head in two flips 0.75 75% 3:1
Birthday match (23 people) 0.507 50.7% ~1:1
UK National Lottery jackpot 0.0000000221 0.0000022% 1:45,057,474

Worked Examples for UK Students

Example 1: Simple Probability (GCSE)

A bag contains 4 red balls, 3 blue balls, and 5 green balls. One ball is picked at random. What is the probability of picking a red ball?
Solution:
Total balls = 4 + 3 + 5 = 12
Favorable outcomes (red) = 4
P(red) = 4/12 = 1/3 โ‰ˆ 0.333 or 33.3%

Example 2: Combinations (A-Level)

A committee of 4 people is to be chosen from 10 candidates. How many different committees are possible?
Solution:
Order doesn't matter, so use combinations
ยนโฐCโ‚„ = 10! / (4! ร— 6!)
= 10 ร— 9 ร— 8 ร— 7 / (4 ร— 3 ร— 2 ร— 1)
= 5040 / 24 = 210 committees

Example 3: Permutations (A-Level)

In a race with 8 runners, how many different ways can the gold, silver, and bronze medals be awarded?
Solution:
Order matters (1st, 2nd, 3rd are different), so use permutations
โธPโ‚ƒ = 8! / (8-3)! = 8! / 5!
= 8 ร— 7 ร— 6 = 336 ways

Example 4: Independent Events (GCSE/A-Level)

A coin is flipped and a die is rolled. What is the probability of getting heads AND rolling a 6?
Solution:
Events are independent
P(heads) = 1/2
P(rolling 6) = 1/6
P(heads AND 6) = 1/2 ร— 1/6 = 1/12 โ‰ˆ 0.0833 or 8.33%

Example 5: Binomial Distribution (A-Level)

A multiple choice test has 10 questions, each with 4 options. If a student guesses randomly, what's the probability of getting exactly 3 correct?
Solution:
n = 10 trials, k = 3 successes, p = 1/4 = 0.25
P(X=3) = ยนโฐCโ‚ƒ ร— (0.25)ยณ ร— (0.75)โท
= 120 ร— 0.015625 ร— 0.1335
โ‰ˆ 0.2503 or 25.03%

Example 6: Conditional Probability (A-Level)

In a class, 60% are girls. 30% of girls and 40% of boys wear glasses. Given that a randomly selected student wears glasses, what's the probability they are a boy?
Solution:
P(Girl) = 0.6, P(Boy) = 0.4
P(Glasses|Girl) = 0.3, P(Glasses|Boy) = 0.4
P(Glasses) = 0.6 ร— 0.3 + 0.4 ร— 0.4 = 0.34
P(Boy|Glasses) = (0.4 ร— 0.4) / 0.34 = 0.471 or 47.1%

Practice Problems

GCSE Easy

A spinner has 8 equal sections numbered 1-8. What is the probability of spinning an odd number?

Answer: 0.5 or 1/2 or 50%
Odd numbers: 1, 3, 5, 7 = 4 outcomes
Total outcomes: 8
P(odd) = 4/8 = 1/2

GCSE Easy

A bag contains 5 red, 3 blue, and 2 yellow marbles. What is the probability of NOT picking a blue marble?

Answer: 0.7 or 7/10 or 70%
P(blue) = 3/10
P(not blue) = 1 - 3/10 = 7/10

GCSE Medium

Two dice are rolled. What is the probability that the sum is 7?

Answer: 1/6 โ‰ˆ 0.167 or 16.7%
Ways to get 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 ways
Total outcomes: 6 ร— 6 = 36
P(sum = 7) = 6/36 = 1/6

A-Level Medium

In how many ways can 5 different books be arranged on a shelf if 2 particular books must be together?

Answer: 48 arrangements
Treat the 2 books as one unit: 4 units to arrange = 4! = 24
The 2 books can swap positions: 2! = 2
Total = 24 ร— 2 = 48

A-Level Hard

A fair coin is tossed 5 times. What is the probability of getting at least 3 heads?

Answer: 0.5 or 1/2 or 50%
P(X โ‰ฅ 3) = P(3) + P(4) + P(5)
= โตCโ‚ƒ(0.5)โต + โตCโ‚„(0.5)โต + โตCโ‚…(0.5)โต
= (10 + 5 + 1) ร— (1/32)
= 16/32 = 1/2

A-Level Hard

A word is formed using all letters of STATISTICS. How many distinguishable arrangements are there?

Answer: 50,400 arrangements
STATISTICS has 10 letters with repeats:
S=3, T=3, A=1, I=2, C=1
Total = 10! / (3! ร— 3! ร— 1! ร— 2! ร— 1!)
= 3,628,800 / (6 ร— 6 ร— 1 ร— 2 ร— 1)
= 3,628,800 / 72 = 50,400

Tips for Probability Success in UK Exams

โœ… Always Draw a Diagram

For complex problems, draw a sample space diagram, tree diagram, or Venn diagram. UK examiners award method marks for clear diagrams, even if your final answer is wrong. Tree diagrams are especially valued in GCSE and A-Level papers.

โœ… Identify the Question Type

Before calculating, identify whether you're dealing with:

  • "And" = Multiplication (events both happening)
  • "Or" = Addition (either event happening)
  • "At least" = Often easier to use 1 - P(none)
  • "Given that" = Conditional probability
  • "How many ways" = Combinations or permutations

โš ๏ธ Common Mistakes to Avoid

  • Confusing combinations (order doesn't matter) with permutations (order matters)
  • Adding probabilities when you should multiply (AND vs OR)
  • Forgetting to subtract P(A โˆฉ B) in the addition rule for non-exclusive events
  • Using "with replacement" calculations when selection is "without replacement"
  • Not reducing fractions to simplest form (examiners may require this)

๐Ÿ“ฑ Calculator Tips for Exams

On scientific calculators approved for UK exams:

  • Combinations: Use the nCr button (often with SHIFT + ร—)
  • Permutations: Use the nPr button (often with SHIFT + รท)
  • Factorial: Use n! button or enter as n ร— (n-1) ร— ... ร— 1
  • Powers: Use ^ or xy for binomial calculations

Probability in UK Exam Boards

Exam Board GCSE Topics A-Level Topics Weighting
AQA Basic probability, tree diagrams, sample spaces, experimental probability Binomial, normal distributions, hypothesis testing, conditional probability 15-20% of statistics content
Edexcel Probability scales, combined events, mutually exclusive events Discrete/continuous distributions, central limit theorem, regression 20-25% of statistics content
OCR Theoretical/experimental probability, Venn diagrams, independent events Probability distributions, sampling, significance testing 15-20% of statistics content
WJEC Probability notation, tree diagrams, relative frequency Statistical distributions, inference, correlation 15-20% of statistics content
SQA (Scotland) Probability and its applications in National 5 Advanced probability in Higher Mathematics Part of statistics unit

Real-World Applications of Probability

๐ŸŽฐ Gaming & Gambling

The UK Gambling Commission requires probability disclosures. Understanding odds helps evaluate risk in betting, lotteries (National Lottery odds: 1 in 45 million for jackpot), and casino games.

๐Ÿฅ Medicine & Health

NHS screening tests use probability to assess disease likelihood. Understanding false positives, false negatives, and Bayesian probability helps interpret medical test results accurately.

๐Ÿ“Š Finance & Insurance

UK insurers calculate premiums using actuarial probability. Investment analysts use probability distributions to model market behaviour and assess portfolio risk.

๐ŸŒค๏ธ Weather Forecasting

The Met Office uses probability to express forecast confidence. "60% chance of rain" means similar conditions produced rain 60% of the time historically.

๐Ÿญ Quality Control

British manufacturers use statistical probability to maintain quality standards, calculating defect rates and determining acceptable sampling methods.

๐Ÿงฌ Genetics

Punnett squares use probability to predict inheritance patterns. Genetic counselling in the NHS relies on probability to advise on hereditary conditions.

๐Ÿ‘จโ€๐Ÿซ

Dr. Eleanor Mitchell, PhD

Chartered Mathematics Educator | PGCE, QTS | Statistics Specialist

Dr. Mitchell is a qualified mathematics teacher with 18 years of experience in UK secondary education. She holds a PhD in Applied Statistics from the University of Cambridge and has authored GCSE and A-Level revision guides. She currently works as an educational consultant, specialising in probability and statistics pedagogy.

Frequently Asked Questions

How do I calculate simple probability? +
To calculate simple probability, divide the number of favorable outcomes by the total number of possible outcomes. For example, the probability of rolling a 3 on a six-sided die is 1/6 (one favorable outcome out of six possible outcomes). Our probability calculator does this automatically - just enter your favorable outcomes and total outcomes.
What is the difference between combinations and permutations? +
Combinations (nCr) count selections where order doesn't matter. For example, choosing 3 students from 10 for a team is a combination - it doesn't matter which order you pick them. Permutations (nPr) count arrangements where order matters. For example, choosing 3 students for 1st, 2nd, and 3rd place is a permutation - the order matters. The formula for combinations is n!/(r!(n-r)!), while permutations use n!/(n-r)!.
How can I use this calculator for GCSE maths revision? +
This probability calculator is perfect for GCSE maths revision. Use the Simple Probability tab for basic probability questions with favorable and total outcomes. The Probability Rules tab helps with combined events (AND, OR, NOT). Work through the practice problems section, which is organised by difficulty level, to build your skills progressively. The worked examples show step-by-step solutions in the format expected by UK exam boards.
What is binomial probability and when do I use it? +
Binomial probability calculates the chance of getting exactly k successes in n independent trials, where each trial has probability p of success. Use it for questions like "What's the probability of getting exactly 3 heads in 10 coin flips?" The formula is C(n,k) ร— p^k ร— (1-p)^(n-k). This is primarily tested at A-Level but understanding the concept helps with higher-level GCSE problems.
How do I convert between probability, percentage, and odds? +
Probability is expressed as a decimal from 0 to 1. To convert to percentage, multiply by 100. For odds, if probability is p, then odds for are p:(1-p). Example: If P = 0.25, then percentage = 25% and odds = 0.25:0.75 = 1:3 (one in four chance). Our calculator automatically shows all three formats for any probability you calculate.
What's the difference between theoretical and experimental probability? +
Theoretical probability is calculated mathematically assuming ideal conditions (e.g., a fair coin has exactly 0.5 probability of heads). Experimental probability (also called empirical or relative frequency) is based on actual observations from experiments. In GCSE, you're expected to understand both and know that experimental probability approaches theoretical probability as the number of trials increases (the Law of Large Numbers).
How do tree diagrams work for calculating probability? +
Tree diagrams visually represent the outcomes of successive events. Each branch shows a possible outcome with its probability. To find the probability of a specific path, multiply along the branches. To find the probability of multiple outcomes, add the relevant path probabilities. They're especially useful for "without replacement" problems and are a key technique in GCSE mathematics.
Can this calculator help with A-Level Further Mathematics? +
Yes! The Binomial tab covers binomial distributions including cumulative probabilities. The Probability Rules tab handles conditional probability calculations needed for Bayes' Theorem questions. The Combinations and Permutations tabs support counting problems. For more advanced topics like Poisson or normal distributions, you would need additional specialised calculators, but this covers the core probability content.

๐Ÿ“š Official UK Resources

Quick Stats

1 in 6
Chance of rolling any number
50%
Coin flip probability
120
Ways to arrange 5 items (5!)

๐ŸŽฏ Quick Reference

  • P(A) range: 0 โ‰ค P(A) โ‰ค 1
  • Impossible: P = 0
  • Certain: P = 1
  • Equal: P = 0.5
  • 0! = 1
  • nC0 = 1
  • nCn = 1

๐Ÿ“– Exam Tip

In UK exams, always show your working! Write the probability formula first, then substitute values. Examiners award method marks even if your final answer is incorrect.

๐Ÿ“š Related Topics