Probability Calculator

📚 How It Works

Factorial (n!)

The factorial of n is the product of all positive integers from 1 to n:

n! = n × (n-1) × (n-2) × ... × 2 × 1

Example: 5! = 5 × 4 × 3 × 2 × 1 = 120

Note: 0! = 1 by definition

Combinations (nCr)

Number of ways to choose r items from n items, order doesn't matter:

C(n,r) = n! / [r! × (n-r)!]

Example: How many 5-card poker hands from 52 cards?

C(52,5) = 52! / (5! × 47!) = 2,598,960

• Use for: Lottery numbers, committees, card hands
• Example: Choosing 3 toppings from 10 → C(10,3) = 120 ways

Permutations (nPr)

Number of ways to arrange r items from n items, order matters:

P(n,r) = n! / (n-r)!

Example: How many ways to award gold, silver, bronze to 8 athletes?

P(8,3) = 8! / 5! = 8 × 7 × 6 = 336

• Use for: Race positions, passwords, seating arrangements
• Example: Arranging 4 books on a shelf → P(4,4) = 4! = 24 ways

Combinations vs. Permutations

Key difference: Does order matter?

• Combination: ABC = BAC = CAB (same group)
• Permutation: ABC ≠ BAC ≠ CAB (different arrangements)

Relationship: P(n,r) = C(n,r) × r!

Basic Probability

Probability of an event occurring:

P(E) = (Number of favorable outcomes) / (Total possible outcomes)

• Probability ranges from 0 (impossible) to 1 (certain)
• Often expressed as percentage (multiply by 100)
• Example: P(rolling a 6) = 1/6 ≈ 0.167 = 16.7%

Multiple Independent Events

Independent: One event doesn't affect the other (e.g., two coin flips)

• P(A AND B): P(A) × P(B) - Both events occur
• P(A OR B): P(A) + P(B) - P(A AND B) - At least one occurs
• P(NOT A): 1 - P(A) - Event doesn't occur

Example: Probability of two heads in two coin flips:

P(H AND H) = 0.5 × 0.5 = 0.25 = 25%

Real-World Applications

• Gambling: Card game odds, lottery chances
• Genetics: Probability of inherited traits
• Quality Control: Defect rates in manufacturing
• Insurance: Risk assessment and pricing
• Sports: Tournament brackets, playoff scenarios
• Cryptography: Password strength, key combinations
• Medicine: Treatment success rates, diagnostic tests

Common Probability Scenarios

Important Notes

• Independence: Events must be independent for AND/OR rules to work
• With/Without Replacement: Affects probability in sequential events
• Mutually Exclusive: Events that can't both happen (e.g., rolling 2 and 6)
• Large numbers: For n > 170, factorial calculations may overflow