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Probability Calculator — Combos & Perms

Calculate probability, combinations, and permutations

At Least Once

99.90%

P(A)

50.00%

Expected

5.00

Results

Probability P(A)
50.00%
Complement P(A')
50.00%
At Least Once in 10 Trials
99.90%
Never in 10 Trials
0.10%
Expected Occurrences
5.00
out of 10 trials

Visual Breakdown

Success Probability50%
Failure Probability50%
At Least Once in 10 Trials100%

Formulas Used

Combinations

C(n,r) = n! / (r! × (n-r)!)

Count the number of ways to choose r items from n items where order does not matter.

Where:

n= Total number of items
r= Number of items chosen

Permutations

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

Count the number of ordered arrangements of r items chosen from n items.

Where:

n= Total number of items
r= Number of items chosen

Complement Rule

P(at least one) = 1 - (1 - p)^n

Calculate the probability of an event occurring at least once over multiple independent trials.

Where:

p= Probability of the event in a single trial
n= Number of independent trials

Example Calculations

1Coin Flip - At Least One Heads in 5 Flips

Inputs

Probability0.5
Trials5

Result

At Least Once96.88%
Complement50.00%
Never in 5 Trials3.13%
Expected Occurrences2.50

P(at least one heads) = 1 - (1 - 0.5)^5 = 1 - 0.03125 = 0.96875 = 96.88%.

2Lottery: Choose 6 from 49

Inputs

n (total)49
r (chosen)6

Result

Combinations C(49,6)13,983,816
Permutations P(49,6)10,068,347,520

C(49,6) = 49! / (6! × 43!) = 13,983,816. This means there are nearly 14 million possible lottery combinations.

Frequently Asked Questions

Q

What is the difference between combinations and permutations?

Combinations count the number of ways to choose items where order does not matter (e.g., picking 3 team members from 10 people). Permutations count arrangements where order matters (e.g., assigning 1st, 2nd, 3rd place). C(10,3) = 120 but P(10,3) = 720, because each combination of 3 can be arranged in 6 different orders.

  • C(10,3) = 120 — choosing 3 people from 10 for a committee (order irrelevant)
  • P(10,3) = 720 — assigning president, VP, secretary from 10 people (order matters)
  • P(n,r) = C(n,r) × r! — each combination has r! arrangements
  • Lottery "pick 6 from 49": C(49,6) = 13,983,816 possible tickets
  • A 4-digit PIN (digits 0–9): P(10,4) = 5,040 if no repeats; 10⁴ = 10,000 with repeats
ScenarioTypeFormulaResult
3 from 10 (committee)CombinationC(10,3)120
3 from 10 (ranked)PermutationP(10,3)720
5 from 52 (poker hand)CombinationC(52,5)2,598,960
6 from 49 (lottery)CombinationC(49,6)13,983,816
Q

How do I calculate the probability of at least one success?

Use the complement rule: P(at least one) = 1 - P(none). For independent events with probability p over n trials: P(at least once) = 1 - (1-p)^n. For example, the chance of rolling at least one 6 in 4 rolls is 1 - (5/6)^4 = 51.77%.

  • Rolling at least one 6 in 4 dice rolls: 1 – (5/6)⁴ = 51.77%
  • Flipping at least one heads in 5 coin flips: 1 – (0.5)⁵ = 96.88%
  • At least one defective in 20 items (2% defect rate): 1 – (0.98)²⁰ = 33.24%
  • At least one birthday match in a group of 23 people: ~50.73% (birthday paradox)
  • This shortcut avoids summing P(1) + P(2) + P(3) + ... individually
Q

What is a complement in probability?

The complement of an event A, written P(A'), is the probability that event A does NOT occur. It equals 1 - P(A). If the probability of rain is 0.30 (30%), the complement (no rain) is 0.70 (70%). The sum of an event and its complement always equals 1.

  • P(rain) = 0.30 means P(no rain) = 0.70 — they always sum to 1.0
  • P(passing an exam) = 0.85 means P(failing) = 0.15
  • Often easier to calculate what you do NOT want, then subtract from 1
  • Example: probability of NOT rolling a 6 on a die = 5/6 ≈ 83.33%
Q

What does C(n,r) mean?

C(n,r) means 'n choose r' — the number of ways to choose r items from n items without regard to order. The formula is C(n,r) = n! / (r! × (n-r)!). For example, C(5,2) = 5! / (2! × 3!) = 120 / (2 × 6) = 10.

  • C(5,2) = 10 — there are 10 ways to pick 2 items from 5
  • C(n,0) = 1 and C(n,n) = 1 — there is always exactly 1 way to choose nothing or everything
  • C(n,r) = C(n, n–r) — choosing 3 from 10 is the same count as choosing 7 from 10
  • Excel/Sheets: use =COMBIN(n, r) to calculate directly
Q

When would I use permutations vs combinations?

Use permutations when the order of selection matters: arranging books on a shelf, assigning ranked positions, creating passwords. Use combinations when order does not matter: choosing team members, selecting lottery numbers, picking items from a menu.

  • Permutation examples: race finishing order, locker combination codes, seating arrangements
  • Combination examples: pizza toppings from a menu, raffle winners, card hands
  • Passwords are permutations — "ABC" and "CBA" are different passwords
  • Team selection is a combination — {Alice, Bob} = {Bob, Alice} is the same team
  • When in doubt: ask "Does rearranging the selection create a different outcome?"

Understanding Probability, Combinations, and Permutations

Probability is the mathematical study of likelihood and uncertainty. It provides tools for quantifying how likely events are to occur, from simple coin flips to complex statistical models. Basic probability concepts apply across statistics, data science, gambling, insurance, and everyday decision-making.

Combinations and permutations are fundamental counting techniques in probability. Combinations (C(n,r)) count the number of ways to select r items from n items when order does not matter. Permutations (P(n,r)) count the arrangements when order matters. The key formula difference is that C(n,r) = P(n,r) / r!, because each combination can be arranged in r! different orders.

The complement rule is one of the most useful tools in probability. Instead of calculating the probability of a complex event directly, it is often easier to calculate the probability that the event does NOT happen (the complement) and subtract from 1. This technique is especially powerful for 'at least one' problems.

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Last Updated: Mar 9, 2026

This calculator is provided for informational and educational purposes only. Results are estimates and should not be considered professional financial, medical, legal, or other advice. Always consult a qualified professional before making important decisions. UseCalcPro is not responsible for any actions taken based on calculator results.

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