The probability of an event is the number of favorable outcomes divided by the total number of outcomes. When the order of a selection matters, count it with permutations, nPr; when it does not, count it with combinations, nCr. Both counts are built out of factorials.
Key Takeaways
- A single die roll lands on a chosen number with probability
1/6≈0.1667.- Two dice sum to 7 in 6 of the 36 outcomes:
6/36≈0.1667.permutations(5,3)=60, butcombinations(5,3)=10: order matters for the first one.- A five-card poker hand:
combinations(52,5)=2,598,960.
Basic probability
Probability is a ratio. Count the outcomes that count as a success, count the total outcomes, and divide the two. The result sits between 0 and 1, where 0 means impossible and 1 means certain.
Worked example 1: dice and cards. A standard die has 6 faces, so the probability of rolling a 4 is 1 ÷ 6, about 0.1667. Type 1/6 → = 0.1667. A 52-card deck has 4 aces, so drawing an ace is 4 ÷ 52 = 1/13, about 0.0769. Type 4/52 → = 0.0769. A third case shows the same ratio landing on a round number: half the deck is red, so a red card has probability 26/52 = 0.5, and 26/52 → = 0.5. All three counts, favorable and total, are whole-number tallies you can verify by hand.
Permutations: when order matters
A permutation is an arrangement where the order counts. A four-digit code, the finishing order of a race, the sequence of songs in a playlist: swap any two items and you have a different result. The count is nPr = n! ÷ (n − r)!, choosing r items from n.
Worked example 2: ordered picks. Pick 3 books from 5 and place them on a shelf. The first slot has 5 choices, the second 4, the third 3, so 5 × 4 × 3 = 60. That is permutations(5,3) → = 60. The same logic arranges all 4 photos in a row: 4! = 24, so permutations(4,4) → = 24. Codes that forbid repeated digits work the same way: a 4-digit code picked from the digits 0 through 9 has 10 × 9 × 8 × 7 = 5,040 options, or permutations(10,4) → = 5,040.
Combinations: when order does not matter
A combination is a group where order is irrelevant. A committee of 3 people is the same committee no matter which order you list the members in. The count is nCr = n! ÷ [r! × (n − r)!], which divides out the r! orderings you do not care about.
Worked example 3: unordered picks. Choose 3 books from 5 and stuff them in a bag. The 60 ordered pickings collapse into groups of 3! = 6 orderings each, leaving 60 ÷ 6 = 10 distinct bags. That is combinations(5,3) → = 10. A committee of 2 from 6 people gives combinations(6,2) → = 15.
How to decide between nPr and nCr
Ask one question: does the result change when two items swap places? If it does, count permutations. If it does not, count combinations. A code 12 differs from 21, so codes are permutations. A hand that is King and 7 of diamonds is the same hand as 7 of diamonds and King, so poker hands are combinations. That single question settles nearly every case, and the factorials explained post covers the n! machinery both formulas rely on.
Compound events and large counts
Some probabilities take a short chain of reasoning instead of one fraction.
Worked example 4: at least one head. Toss three coins and ask for at least one head. Eight outcomes are possible, and only one of them, three tails, fails. So the probability is 7 ÷ 8 = 0.875. Type 7/8 → = 0.875. When the counts get large the ratio still works, it just hides inside enormous numbers: matching 6 numbers on a 49-ball lottery has one winning combination out of combinations(49,6) = 13,983,816. Stating that as a percent takes the same favorable ÷ total × 100 conversion the percentage calculator handles. Where probability predicts what a long run of trials looks like, the statistics calculator summarizes the observed data instead.
Frequently asked questions
What do nPr and nCr stand for?
nPr counts the permutations of r items taken from n, where order matters. nCr counts the combinations, where order does not. Both read as choosing r from n.
Why is nPr always larger than nCr?
Because every combination of r items can be listed in r! different orders. Multiplying the combination count by r! gives the permutation count, so for the same n and r, permutations come out bigger. For 5 and 3 the numbers are 60 and 10.
Is a probability ever greater than 1?
No. The number of favorable outcomes can never exceed the total number of outcomes, so the ratio stays between 0 and 1. A probability of exactly 1 means the event is certain.
How do I calculate permutations and combinations on the calculator?
Use the permutations(n, r) and combinations(n, r) functions, or the nPr and nCr entries on the factorial calculator. For example, combinations(52,5) → = 2,598,960.