What is the difference between combinations and permutations?

Combinations count selections where order doesn't matter: C(n,r) = n! / (r!(n-r)!). Permutations count arrangements where order matters: P(n,r) = n! / (n-r)!. For example, choosing 3 people from 10 for a committee is C(10,3) = 120. Arranging 3 people in 1st, 2nd, 3rd place from 10 is P(10,3) = 720.

How do you calculate P(A AND B)?

For independent events, P(A AND B) = P(A) × P(B). For example, the probability of rolling a 6 on a die AND flipping heads on a coin = (1/6) × (1/2) = 1/12. For dependent events, P(A AND B) = P(A) × P(B|A).

How do you calculate P(A OR B)?

P(A OR B) = P(A) + P(B) - P(A AND B). The subtraction prevents double-counting when both events can occur. For mutually exclusive events, P(A AND B) = 0, so P(A OR B) = P(A) + P(B).

Probability Calculator

Single Event Probability P(A)

Favorable Outcomes

Total Outcomes

P(A)

1/6 = 0.1667 = 16.67%

Multiple Events (Independent)

P(A) as decimal (0-1)

P(B) as decimal (0-1)

P(A AND B) = P(A) × P(B)

0.15 = 15%

P(A OR B) = P(A)+P(B)−P(A∩B)

0.65 = 65%

P(not A)

0.5 = 50%

P(not B)

0.7 = 70%

Combinations & Permutations

n (total items)

r (chosen items)

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

120

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

720

Understanding Probability, Combinations, and Permutations

Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). The basic formula is P(A) = favorable outcomes divided by total possible outcomes. For example, the probability of rolling a 3 on a standard die is 1/6, since there is one favorable outcome out of six possibilities.

When working with multiple events, two key rules apply. For independent events happening together (AND), multiply their individual probabilities: P(A ∩ B) = P(A) × P(B). For the probability of either event occurring (OR), add their probabilities and subtract the overlap: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The complement rule states P(not A) = 1 − P(A).

Combinations and permutations count the ways to select items from a group. Combinations ignore order: C(n,r) = n! / (r!(n-r)!). Use combinations when choosing a committee or lottery numbers. Permutations consider order: P(n,r) = n! / (n-r)!. Use permutations when arranging items in specific positions, like race placements or password characters. This calculator handles all these calculations, displaying results as fractions, decimals, and percentages for maximum clarity.

Formula

Basic Probability:

P(A) = Favorable Outcomes / Total Outcomes

Union (OR):

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Intersection (AND) for independent events:

P(A ∩ B) = P(A) × P(B)

Conditional Probability:

P(A | B) = P(A ∩ B) / P(B)

Where:

P(A) = probability of event A occurring
P(A ∪ B) = probability of A or B (or both) occurring
P(A ∩ B) = probability of both A and B occurring
P(A | B) = probability of A given that B has occurred

Example Calculation

Scenario: What is the probability of rolling a 5 OR getting heads on a coin?

Step 1: P(rolling 5) = 1/6 ≈ 0.167
Step 2: P(heads) = 1/2 = 0.5
Step 3: Events are independent, so P(A ∩ B) = 1/6 × 1/2 = 1/12 ≈ 0.083
Step 4: P(A ∪ B) = 0.167 + 0.5 − 0.083 = 0.583
Result: 58.3% chance of rolling a 5 or flipping heads (or both)