Probability Calculator
Calculate combinations, permutations, binomial probability, and more with step-by-step solutions. 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)
Key Formulas
Factorial
n! = n × (n-1) × ... × 2 × 1
5! = 5 × 4 × 3 × 2 × 1 = 120
Permutations
nPr = n! / (n-r)!
5P2 = 5!/(5-2)! = 120/6 = 20
Combinations
nCr = n! / (r! × (n-r)!)
5C2 = 5!/(2! × 3!) = 120/12 = 10
Basic Probability
P(E) = Favorable / Total
P(Ace) = 4/52 = 1/13
Addition Rule
P(A∪B) = P(A) + P(B) - P(A∩B)
P(A∪B) = 0.3 + 0.4 - 0.1 = 0.6
Conditional
P(A|B) = P(A∩B) / P(B)
P(A|B) = 0.12 / 0.4 = 0.3
Common Mistakes
Confusing nPr with nCr
nPr is for arrangements (order matters), nCr is for selections (order does not matter). nPr is always ≥ nCr.
Forgetting 0! = 1
By definition, 0! = 1, not 0. This is needed for formulas like nC0 = 1.
Adding P(A∩B) instead of subtracting
In P(A∪B) = P(A) + P(B) - P(A∩B), we subtract P(A∩B) because it is counted twice.
Using addition for independent events
For independent events, P(A∩B) = P(A) × P(B), not P(A) + P(B).
Confusing P(A|B) with P(B|A)
P(A|B) ≠ P(B|A) unless A and B have special relationships. Always check what is given.
Probability > 1 or < 0
All probabilities must be between 0 and 1. If you get a value outside this range, check your calculation.
Understanding Probability & Combinatorics
What is Probability?
Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain).
Key formula:
P(Event) = Favorable outcomes / Total outcomes
Example: P(rolling 6) = 1/6
Permutations vs Combinations
- nPr:Order MATTERS (arrangements)
- nCr:Order does NOT matter (selections)
- •nPr is always ≥ nCr
- •nCr = nPr ÷ r!
Worked Examples
Example 1: Combinations (nCr)
How many ways can you choose 3 people from a group of 10 for a committee?
Step 1: Identify: n = 10, r = 3
Step 2: Formula: 10C3 = 10! / (3! × 7!)
Step 3: Calculate: 3,628,800 / (6 × 5,040)
Result: 10C3 = 120 ways
Example 2: Permutations (nPr)
How many ways can 8 runners finish 1st, 2nd, and 3rd place?
Step 1: Identify: n = 8, r = 3
Step 2: Formula: 8P3 = 8! / (8-3)! = 8! / 5!
Step 3: Simplify: 8 × 7 × 6 = 336
Result: 8P3 = 336 arrangements
Example 3: Binomial Probability
What is the probability of getting exactly 3 heads in 5 coin flips?
Step 1: n = 5, k = 3, p = 0.5
Step 2: P(X=3) = 5C3 × 0.5³ × 0.5²
Step 3: = 10 × 0.125 × 0.25
Result: P(X=3) = 0.3125 = 31.25%
Example 4: Conditional Probability
If P(A∩B) = 0.15 and P(B) = 0.5, what is P(A|B)?
Step 1: Formula: P(A|B) = P(A∩B) / P(B)
Step 2: Substitute: P(A|B) = 0.15 / 0.5
Result: P(A|B) = 0.3 = 30%
Exam Tips
Order Matters?
Ask yourself: does the order of selection matter? If yes → nPr. If no → 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!
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, check your work!
Complement Trick
P(at least one) = 1 - P(none). Often easier than calculating directly!
Frequently Asked Questions
What is the difference between permutations and combinations?
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How do I know when to use nPr vs nCr?
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What is conditional probability P(A|B)?
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When do I use the binomial probability formula?
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Why is 0! equal to 1?
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