Start with the recap, study the fully worked examples, then use the practice problems to
check your understanding of Sample Space.
This page combines explanation, solved examples, and follow-up practice so you can move
from recognition to confident problem-solving in Statistics.
Concept Recap
The sample space is the complete set of all possible outcomes for a probability experiment, listed without repetition. It forms the foundation for every probability calculation because the probability of any event is a fraction of the sample space.
Before calculating probability, list every possible outcome. For a die: {1,2,3,4,5,6}. For two coins: {HH,HT,TH,TT}. That's your sample space - the complete menu of what could happen.
Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:Sample Space starts by naming the possible outcomes and the event rule before assigning or combining probabilities.
Common stuck point:Students often know a procedure related to sample space but skip the recognition step: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition? That leads to a calculation or graph that looks reasonable but answers a different question.
Sense of Study hint:Ask: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?
Worked Examples
Example 1
medium
List the sample space for flipping three coins (H, T) and count its size.All 8 outcomes in the sample space for three coin flips
Answer
8 outcomes
First step
1
{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}.
See the full worked solution + why-it-works coaching
Setup·Key insight·Why it works·Common pitfall·Connection
Two dice are rolled. How many outcomes give a product equal to 12?
Example 3
hard
A 3-letter code uses letters from {A, B, C, D} with repetition allowed. How many codes are in the sample space, and how many use no repeated letter?
Example 4
challenge
Two cards are drawn from a 52-card deck without replacement and order does not matter. How many outcomes are in the sample space?
Example 5
challenge
A die is rolled until a 6 appears or until 3 rolls have occurred (whichever comes first). How many distinct sequences are in the sample space?
Example 6
easy
List the sample space for rolling a standard six-sided die and flipping a coin simultaneously.
Example 7
medium
A restaurant offers 3 starters (soup, salad, bread), 4 mains (chicken, fish, beef, pasta), and 2 desserts (cake, fruit). How many different 3-course meals are possible? Do you need to list them all?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
List the sample space for rolling a single standard die.
Example 2
easy
How many outcomes are in the sample space for flipping two coins?Sample space for flipping two coins
Example 3
easy
List the sample space for flipping a single coin.
Example 4
easy
How many outcomes are in the sample space when rolling two dice (ordered)?
Example 5
easy
List the sample space for the sum when rolling two dice.
Example 6
easy
A spinner has colors red, blue, green, yellow. What is its sample space?
Example 7
easy
How many outcomes are in the sample space for flipping three coins?Sample space for flipping three coins
Example 8
easy
List the sample space for drawing one card and noting its suit.
Example 9
medium
How many outcomes are in the sample space for flipping a coin and then rolling a die?Sample space for flipping a coin then rolling a die
Example 10
medium
Two dice are rolled. How many outcomes in the sample space give a sum of 5?
Example 11
medium
From the menu of one appetizer (2 choices) and one main (3 choices), how many meal outcomes are possible?
Example 12
medium
A coin is flipped twice. List the sample space where order matters.All ordered outcomes for two coin flips
Example 13
medium
A spinner (1-3) is spun and a coin flipped. How many outcomes are in the sample space?Sample space for spinning a 3-section spinner then flipping a coin
Example 14
medium
Two distinct letters are chosen in order from {A, B, C}. How many outcomes are in the sample space?
Example 15
medium
A bag has marbles labeled A, B, C, D. Two are drawn without replacement, order ignored. How many outcomes?
Example 16
medium
An outfit is one of 3 shirts and one of 4 pants. How many outfits are in the sample space?
Example 17
medium
A 4-digit PIN uses digits 0-9 with repeats allowed. How many PINs are in the sample space?
Example 18
challenge
Two dice are rolled. How many outcomes in the sample space have at least one die showing a 6?Outcomes where at least one die shows a 6 (11 total)
Example 19
challenge
How many outcomes are in the sample space for arranging the letters A, B, C in a row?
Example 20
challenge
A coin is flipped until the first head appears, up to 3 flips. List the sample space.Sample space: flip until first head or 3 flips
Example 21
easy
List the sample space for drawing one ball from a bag with balls labeled A, B, C, D, E.
Example 22
easy
How many outcomes are in the sample space for flipping a coin and then drawing a card from a standard 52-card deck?
Example 23
easy
A spinner has 10 equal sections numbered 1 through 10. How many outcomes are in the sample space?
Example 24
easy
List the sample space for flipping a coin and rolling a die (use H/T and 1-6).Complete sample space for coin flip followed by die roll
Example 25
easy
A bag contains 3 marbles labeled 1, 2, 3. Two marbles are drawn one after the other without replacement. How many outcomes (ordered pairs) are in the sample space?
Example 26
medium
Two dice are rolled. How many outcomes in the sample space have a sum of 8?
Example 27
medium
A 4-digit PIN uses digits 0-9 with repetition allowed. How many possible PINs are in the sample space?
Example 28
medium
An outfit picks one of 3 shirts and one of 4 pants. How many outfit outcomes are in the sample space?
Example 29
medium
Two dice are rolled and we record only the unordered pair (a, b) with a≤b. How many unordered outcomes are in the sample space?
Example 30
medium
A bag holds 2 red and 2 blue marbles. Two marbles are drawn without replacement; order matters. How many outcomes are in the sample space?
Example 31
medium
A coin is flipped, then a 4-sided die is rolled. How many outcomes are in the sample space?Sample space for flipping a coin then rolling a 4-sided die
Example 32
hard
Three students are seated in a row. How many seating outcomes (sample space) are possible?
Example 33
hard
A password is exactly 5 characters chosen from 26 lowercase letters with no repetition. How many passwords are in the sample space?
Example 34
hard
Two dice are rolled. How many outcomes have an even sum?Even-sum outcomes = both even (intersection, 9) + both odd (outside, 9) = 18
Example 35
hard
A 5-question true/false quiz is answered. How many possible answer sheets are in the sample space?
Example 36
hard
A committee chooses a chair and a secretary from 8 people (different people for each role). How many sample-space outcomes are there?
Example 37
medium
A password consists of one letter (A–E) followed by one digit (1–3). (a) List the entire sample space. (b) How many passwords contain the letter 'C'? (c) What is the probability of randomly generating a password that starts with 'C'?
Example 38
hard
Two dice are rolled. (a) How many outcomes are in the sample space? (b) How many outcomes give a sum of 7? (c) How many outcomes give a sum greater than 10? (d) Which sum is more likely: 7 or greater than 10?