Maths
What is a sample space in probability?
Quick answer
A sample space is the list of every possible outcome of an experiment. Flipping a coin has the sample space {H, T}; rolling a dice has {1, 2, 3, 4, 5, 6}.
In one sentence
The sample space is the complete list of everything that could happen in an experiment, such as {1, 2, 3, 4, 5, 6} for rolling a dice.
Explained
In probability, an experiment is anything with an uncertain result: flipping a coin, rolling a dice, picking a sweet from a bag, choosing a meal from a menu. Each possible result is an outcome, and the set of all possible outcomes is the sample space. It is usually written S (or the Greek letter Ω, omega) with the outcomes inside curly brackets:
- Flip a coin: S = {H, T}
- Roll a dice: S = {1, 2, 3, 4, 5, 6}
- Pick a day of the week: S = {Mon, Tue, Wed, Thu, Fri, Sat, Sun}
The number of outcomes is written n(S). For a dice, n(S) = 6.
An event is a part of the sample space: one or more outcomes you are interested in. "Rolling an even number" is the event A = {2, 4, 6}. When all the outcomes are equally likely, the probability of an event is
P(A) = n(A) ÷ n(S) = number of outcomes in the event ÷ number of outcomes in the sample space
so P(even) = 3/6 = 1/2. That is why the sample space matters: it is the bottom of the fraction. Get it wrong and every probability is wrong.
bag
pizza
juice
With two or more steps, list the outcomes in order. Two coins give HH, HT, TH and TT: four outcomes, not three, because "heads then tails" and "tails then heads" are different. A quick count uses the counting principle: if the first step has m outcomes and the second has n, together there are m × n. Two coins: 2 × 2 = 4. Three coins: 2 × 2 × 2 = 8. Two dice: 6 × 6 = 36.
There are three good ways to show a sample space:
- A list, written systematically so nothing is missed (all the H… outcomes first, then all the T… ones).
- A table (a sample space diagram), perfect for two steps such as two dice or a coin and a spinner.
- A tree diagram, best for three or more steps, such as three coins.
A sample space diagram also shows independent events: events where one has no effect on the other, like the results on two different dice. For independent events, P(A and B) = P(A) × P(B). In the two-dice table, "the first dice is even" covers 18 squares and "the second dice is a 6" covers 6 squares; only 3 squares have both, and 3/36 = 1/12 = 1/2 × 1/6. Do not confuse this with mutually exclusive events, which share no outcomes at all and so can never happen together ("roll a 2" and "roll an odd number"). In a Venn diagram, the rectangle is the sample space and each circle is an event: mutually exclusive circles do not overlap.
Examples
| Experiment | Sample space | n(S) |
|---|---|---|
| Flip a coin | {H, T} | 2 |
| Roll a dice | {1, 2, 3, 4, 5, 6} | 6 |
| Flip two coins | {HH, HT, TH, TT} | 4 |
| Flip three coins | {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} | 8 |
| Spin A, B or C and flip a coin | {AH, AT, BH, BT, CH, CT} | 6 |
| Roll two dice | all pairs (1, 1), (1, 2), … , (6, 6) | 36 |
Sample space diagram for the total of two dice (first dice down the side, second along the top):
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Now just count squares. A total of 7 appears 6 times, so P(7) = 6/36 = 1/6, the most likely total. A total of 12 appears once: P(12) = 1/36. A total of 10 appears 3 times: P(10) = 3/36 = 1/12.
Common mistakes
- Treating HT and TH as one outcome. Then the list {HH, one of each, TT} has three outcomes that are not equally likely. Fix: label the coins (first, second) and keep the order.
- Missing outcomes. Fix: list systematically, then check the count with the counting principle (three coins must give 8).
- Using the totals {2, 3, … , 12} as the sample space for two dice and calling them equally likely. Fix: the 36 pairs are equally likely; the 11 totals are not.
- Mixing up independent and mutually exclusive. Fix: independent means "no influence"; mutually exclusive means "cannot both happen".
Step by step
Step by step
- Describe one outcomeDecide what a single result looks like, such as a pair (first dice, second dice).
- Choose a methodA list for one step, a table for two steps, a tree diagram for three or more.
- List systematicallyWork in order so nothing is missed or repeated.
- Check the countMultiply the number of outcomes for each step; it should match n(S).
- Count the eventIf the outcomes are equally likely, P(event) = n(event) ÷ n(S).
Practice questions
Sample spaces
Try each one, then check your answer.
- Write the sample space for rolling one dice.
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{1, 2, 3, 4, 5, 6} - A spinner is red, blue or green. Write its sample space.
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{red, blue, green} - List the sample space for flipping two coins.
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{HH, HT, TH, TT}: 4 outcomes - Two coins are flipped. What is the probability of exactly one head?
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HT and TH: 2/4 = 1/2 - How many outcomes are there when three coins are flipped?
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2 × 2 × 2 = 8 - Three coins are flipped. What is the probability of exactly two heads?
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HHT, HTH, THH: 3/8 - Three coins are flipped. What is the probability of at least one tail?
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Only HHH has no tail, so 1 − 1/8 = 7/8 - How many outcomes are there when two dice are rolled?
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6 × 6 = 36 - Two dice are rolled. What is the probability of a total of 8?
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(2,6), (3,5), (4,4), (5,3), (6,2): 5/36 - Two dice are rolled. What is the probability of a double?
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(1,1) to (6,6): 6/36 = 1/6 - Two dice are rolled. What is the probability of a total greater than 9?
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Totals 10, 11, 12: 3 + 2 + 1 = 6 outcomes, 6/36 = 1/6 - A coin is flipped and a dice is rolled. How many outcomes are there?
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2 × 6 = 12 - A coin is flipped and a dice is rolled. What is the probability of heads and an even number?
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H2, H4, H6: 3/12 = 1/4 (= 1/2 × 1/2, as the events are independent) - On one dice, are "roll a 2" and "roll an odd number" mutually exclusive?
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Yes: 2 is not odd, so they share no outcomes. - A bag holds two red sweets (R1, R2) and one blue (B). Two are taken, one after the other, without putting any back. List the sample space and find P(both red).
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R1R2, R1B, R2R1, R2B, BR1, BR2: 6 outcomes, 2 of them both red, so 1/3 - Two dice: A = "the first dice shows 6", B = "the total is 7". Are A and B independent?
Show answer
P(A) = 1/6, P(B) = 1/6, and only (6,1) is in both: 1/36 = 1/6 × 1/6, so yes
Word problems
Sample spaces in real life
Try each one, then check your answer.
- A café offers pasta, curry or salad, with water or juice. How many different meals with one main and one drink are there?
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3 × 2 = 6 meals - A padlock has 3 wheels, each with the digits 0 to 9. How many different codes are there?
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10 × 10 × 10 = 1,000 codes - An ice-cream stall has 4 flavours, 2 kinds of cone, and sprinkles or no sprinkles. How many different ice creams can you order?
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4 × 2 × 2 = 16 - Two friends play rock, paper, scissors, each choosing at random. What is the probability of a draw?
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3 × 3 = 9 outcomes, 3 are draws: 1/3 - In a board game you need at least one 6 when you roll two dice. What is the probability?
Show answer
25 of the 36 outcomes have no 6, so 36 − 25 = 11: 11/36
Questions people ask
What is the sample space of rolling two dice?
It is the 36 ordered pairs (1, 1), (1, 2), … , (6, 6). Draw it as a 6-by-6 table. The totals 2 to 12 are not the sample space, because they are not equally likely.
What is the sample space for tossing 3 coins?
{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}: 8 outcomes, because 2 × 2 × 2 = 8. A tree diagram is the easiest way to list them.
What is the symbol for sample space?
Usually S, or Ω (omega) in more advanced work. The outcomes go in curly brackets, and n(S) means the number of outcomes.
What is the difference between a sample space and an event?
The sample space is every possible outcome; an event is a part of it. For one dice, S = {1, 2, 3, 4, 5, 6} and the event "less than 3" is {1, 2}.
Is sample space the same as sample size?
No. Sample space is a probability idea: all possible outcomes. Sample size is a statistics idea: how many people or items are in a sample, such as 50 people in a survey.
What are independent events, and what is the formula?
Events are independent when one does not affect the chance of the other, like two coin flips. Then P(A and B) = P(A) × P(B). If that equation fails, the events are dependent; see conditional probability.
What is the difference between independent and mutually exclusive events?
Mutually exclusive events cannot happen at the same time, so P(A and B) = 0. Independent events can happen together; one just doesn't change the other's chance. Two events that each have a non-zero probability cannot be both.
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Sample space
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