Sample spaces, outcomes and events
A number cube is about to be rolled. “It shows a 4” and “it shows an even number” might both come true, but they are not the same kind of statement. What is the difference, and how is each one written down?
Where you are 14 in Basic probability
Why this exists
Before a number cube is rolled, you can already ask questions about the roll. Will it show a 4? Will it show an even number? After the roll, one look at the top face answers both.
The questions are not alike, though. “A 4” names exactly one face. “An even number” names three faces, 2, 4 and 6, and comes true whichever of them turns up. A roll of 6 is one thing that happened, yet it makes “even” and “more than 4” true at the same time.
Probability will soon put a number on questions like these. First two things must be settled exactly: what are all the things that can happen, and which of them does a question count? The words for these are outcome, sample space and event. The lesson on what randomness means described random processes; this one names their pieces.
Definition
An experiment is an action whose result is not known until it is carried out, such as rolling a number cube, spinning a spinner or drawing one card from a pile. Each time it is carried out is a trial.
An outcome is one single result of one trial. The cube shows 4; the spinner lands on blue; the card drawn is the 7.
The sample space of an experiment is the set of all its possible outcomes. It is usually called S. A set is written by listing its members between curly brackets, called braces, with commas between them; this is roster notation. For one roll of a number cube,
A list counts as a sample space only when both of these hold:
- every possible result of a trial is on the list, and
- every trial gives exactly one member of the list, never two at once.
Each outcome is listed once, and the order does not matter: \{6, 5, 4, 3, 2, 1\} is the same set as \{1, 2, 3, 4, 5, 6\}, because a set is decided entirely by which members belong to it.
An event is any collection of outcomes from the sample space, usually the ones that fit some description you care about. An event is named with a capital letter and written as a set:
Every member of an event is a member of S, so an event is a subset of the sample space, written E \subseteq S. An event happens on a trial when the outcome of that trial is one of its members. Roll a 4: since 4 \in E, the event E happened. Roll a 5: since 5 \notin E, it did not.
Three kinds of event have names of their own.
- An event with exactly one outcome, such as \{4\}, is a simple event.
- The event with no outcomes at all is the impossible event. It is written \{\,\}, a pair of braces with nothing inside, or \varnothing, and is called the empty set. “The cube shows a 9” is the impossible event: no trial can make it happen.
- The whole sample space S is the certain event. “The cube shows a number from 1 to 6” is the certain event: every trial makes it happen.
These two are the ends of the scale of chance, where the words “impossible” and “certain” come from.
Intuition
Spin the spinner in Figure 1. The arrow stops in exactly one of the four sections, so one spin gives one outcome, and the sample space is
Now ask a question about the spin: did it land on red or yellow? Some outcomes answer yes and the rest answer no. The event is the list of outcomes that answer yes, \{\text{red}, \text{yellow}\}. That is the whole difference between the two ideas:
An outcome is what happens. An event is a question about what happens, written as the set of outcomes that answer yes.
A picture helps. Draw the sample space as a frame holding its outcomes, and draw an event as a loop around the outcomes it contains. Figure 2 does this for one roll of a number cube, with event A, “less than 3”, and event B, “more than 3”.
A trial picks one outcome from the frame, and every loop around it has happened. Roll a 5: B happened and A did not. Roll a 3: neither happened, though the roll still gave an outcome.
An outcome and the simple event holding it are close but not the same. The number 4 is a result; the set \{4\} is the event “the cube shows a 4”. Everyday speech calls both “rolling a 4”; the braces say which is meant.
Results
One outcome per trial. Every trial of an experiment gives exactly one outcome in its sample space S.
When an event happens. An event E happens on a trial exactly when that trial’s outcome is a member of E.
The two extremes. The impossible event \varnothing never happens, and the certain event S happens on every trial.
Same members, same event. Two descriptions name the same event when they pick out the same outcomes, however different their words. On one roll of a number cube, “more than 4” and “a 5 or a 6” are the same event, \{5, 6\}.
Why it is true
The first condition on a sample space puts every result on the list, so at least one outcome occurs; the second rules out two at once, so at most one occurs. The second result is the definition of an event happening. Every outcome is a member of S, so S always happens, and \varnothing has no members, so it never does. Last, an event is the set of outcomes it contains, not the words describing it, and two sets with the same members are the same set.
Worked examples
Naming the parts of a spin
The spinner in Figure 1 is spun once and the color is noted.
(a) Name the experiment and one possible outcome. (b) Write the sample space. (c) Write as sets the event A, “lands on blue”, and the event B, “lands on a color whose name has at most four letters”. (d) The arrow stops on red. Which of A and B happened?
Solution(a) The experiment is spinning the spinner once and noting the color. Green is one possible outcome.
(b) There are four sections and the arrow stops in exactly one of them, so S = \{\text{red}, \text{blue}, \text{green}, \text{yellow}\}.
(c) Only blue fits “lands on blue”, so A = \{\text{blue}\}, a simple event. For B, count the letters in each name: red has 3, blue 4, green 5 and yellow 6. “At most four” includes four, so B = \{\text{red}, \text{blue}\}.
(d) Red is not a member of A, so A did not happen. Red is a member of B, so B did happen. One outcome, and the two events give different answers.
Many events from one roll
A number cube is rolled once, so S = \{1, 2, 3, 4, 5, 6\}. Write each event as a set, and say which are simple, which impossible and which certain.
C: the cube shows a multiple of 3. D: it shows more than 4. E: it shows an even number less than 3. F: it shows a number greater than 6. G: it shows a number less than 10.
Then say which of the events happen when the roll is a 6, and which when it is a 1.
SolutionGo through S and keep the outcomes that fit each description.
- The multiples of 3 in S are 3 and 6, so C = \{3, 6\}.
- The outcomes more than 4 are 5 and 6, so D = \{5, 6\}.
- The even outcomes are 2, 4 and 6, and only 2 is less than 3, so E = \{2\}, a simple event.
- No face shows more than 6, so F = \{\,\} = \varnothing, the impossible event.
- Every face shows less than 10, so G = \{1, 2, 3, 4, 5, 6\} = S, the certain event.
A roll of 6 is a member of C, D and G, so those three happen and E and F do not. A roll of 1 is a member of G only. However the cube lands, G happens and F does not.
Two events that share an outcome
One card is drawn from a pile of cards numbered 1 to 12. Event M is “the number is a multiple of 4”, and event H is “the number is at least 9”.
(a) Write S, M and H. (b) Which outcomes make both M and H happen? (c) Which make at least one of them happen? (d) Which make neither happen?
Solution(a) Every card is a possible draw, and each draw is one card:
The dots mean the list carries on in the same pattern up to 12. The multiples of 4 up to 12 are 4, 8 and 12, and “at least 9” includes 9 itself:
(b) Both happen only when the card is in both sets. The only number in both is 12, where the loops in Figure 3 overlap.
(c) At least one happens when the card is in either set. Put the members of both together, writing 12 once: \{4, 8, 9, 10, 11, 12\}.
(d) Neither happens for the cards in no loop: \{1, 2, 3, 5, 6, 7\}.
Check. Parts (c) and (d) between them should cover all 12 cards exactly once, and 6 + 6 = 12. Combining events like this is the subject of union and intersection of events.
What is recorded decides the sample space
A jar holds six buttons: three round, two square and one shaped like a star. One button is taken out without looking and its shape is noted.
(a) Write the sample space. (b) Write the event K, “a shape with corners”, and the event T, “a shape found on more than one button in the jar”. (c) Should S be written with six members, one for each button?
Solution(a) The result noted is a shape, and three different shapes can turn up:
(b) Squares and stars have corners, so K = \{\text{square}, \text{star}\}. Three buttons are round and two are square, but only one is a star, so T = \{\text{round}, \text{square}\}.
(c) No. The outcome is the shape, and any of the three round buttons gives the same result, round. A set lists each member once, so S has 3 outcomes though there are 6 buttons. If the buttons were numbered 1 to 6 and the number noted instead, S would have 6 outcomes. The sample space depends on what is recorded, not only on what is done.
Round does turn up more often than star, but the sample space does not show that. Attaching a likelihood to each outcome is what a probability model adds.
The same words, a different sample space
A parcel is on its way, and the day of the week it arrives is noted, written as Mon, Tue, Wed, Thu, Fri, Sat or Sun. Courier 1 delivers every day; Courier 2 delivers Monday to Friday only. For each courier, write S and the events W, “arrives at the weekend”, and D, “arrives on a weekday”, and say whether either is impossible or certain.
SolutionCourier 1. Any of the seven days is possible, so S holds all seven. Then W = \{\text{Sat}, \text{Sun}\} and D = \{\text{Mon}, \text{Tue}, \text{Wed}, \text{Thu}, \text{Fri}\}. Neither is empty and neither is all of S, so neither is impossible or certain.
Courier 2. Only weekdays are possible:
No outcome in this S is a weekend day, so W = \varnothing, the impossible event. Every outcome is a weekday, so D = S, the certain event.
The words describing W and D did not change; the sample space did. Whether an event is impossible or certain is always a statement about a particular sample space.
Boundaries
A sample space must be complete and must not overlap. \{1, 2, 3, 4, 5\} is not a sample space for one roll of a cube, because a roll of 6 is missing. \{\text{even}, \text{odd}, 6\} is not one either, because a roll of 6 would give two of its members at once.
An outcome is the result of one whole trial. If the experiment is rolling a cube twice, a single outcome is the pair of results, such as 6 then 2, not one number. Writing out such a sample space so that nothing is missed is listing a sample space, and events across several stages are compound events.
A sample space says nothing about how likely its outcomes are. In Example 4, round and star each appear once in S, though round is on three buttons and star on one.
An event needs a description that settles every outcome. “The card shows a big number” is not an event until “big” is pinned down, for instance as “at least 9”.
Not every sample space can be listed. The exact moment a parcel arrives could be any instant in the day, too many to write between braces. Such experiments use lengths and areas instead, as in geometric probability.
Where this goes next
With outcomes and events named, a probability can be given a precise meaning. When the outcomes are equally likely, the probability of an event compares how many outcomes it holds with how many are in S. Probabilities can also be estimated from real trials, in experimental probability. Larger experiments need lists and tables, and events built from other events: “not A” in complements, “A and B” or “A or B” in union and intersection.
Exercises
Sample spaces
- A drawer holds eight socks: three gray, three black and two white. One sock is taken out without looking and its color is noted. Write the sample space.
- A number cube is rolled, and the result is noted only as “odd” or “even”. Write the sample space.
- One card is drawn from cards numbered 1 to 12, and the number of digits on the card is noted. Write the sample space.
- Is \{1, 2, 3, 4, 5\} a sample space for one roll of a number cube? Explain.
- Is \{\text{red}, \text{blue}, \text{green}, \text{yellow}, \text{red or blue}\} a sample space for one spin of the spinner in Figure 1? Explain.
Writing events
For Exercises 6 and 7, a number cube is rolled once. Write each event as a set.
- The cube shows a number that is at most 3.
- The cube shows a factor of 12.
For Exercises 8 to 10, one card is drawn from cards numbered 1 to 12. Write each event as a set.
- The number is prime.
- The number is a square number.
- The number is greater than 12.
- A parcel’s day of arrival is noted, and it can arrive on any day of the week. Write the event “it arrives on a day whose name contains the letter u”.
- The spinner in Figure 1 is spun once. Write the event “it lands on a color other than blue”.
Which events happen
- One card is drawn from cards numbered 1 to 12, and it is the 6. Which of these events happened? P: the number is even. Q: the number is greater than 6. R: the number is a multiple of 3. T: the number is a square number.
- A number cube is rolled once. E = \{1, 2\}, F = \{2, 4, 6\} and G = \{5\}. Name an outcome that makes both E and F happen, and every outcome that makes none of the three happen. Explain why no roll makes both E and G happen.
- A sock is taken from the drawer in Exercise 1. List every event of this experiment, from the impossible event to the certain event, and say which are simple. How many events are there?
Reasoning
- Marisa says that “rolling a 5” is an outcome. Nathaniel says it is an event. Explain how both can be right, and how braces settle which one is meant.
- Explain why every trial makes the certain event happen and no trial makes the impossible event happen.
- On one roll of a number cube, Ottoline describes event A as “an odd number greater than 1” and Percival describes event B as “a prime number other than 2”. Are A and B the same event? Explain.
Answers
- S = \{\text{gray}, \text{black}, \text{white}\}. There are eight socks but only three colors, and each is listed once.
- S = \{\text{odd}, \text{even}\}. Every roll is exactly one of the two.
- S = \{1, 2\}. The cards 1 to 9 have one digit and the cards 10, 11 and 12 have two.
- No. A roll of 6 is possible and is not on the list, so the list does not hold every possible result.
- No. A spin that lands on red would give two members of the list at once, red and “red or blue”. “Red or blue” is an event, \{\text{red}, \text{blue}\}, not a single outcome.
- \{1, 2, 3\}. “At most 3” includes 3.
- \{1, 2, 3, 4, 6\}. The factors of 12 are 1, 2, 3, 4, 6 and 12, and 12 is not on a cube.
- \{2, 3, 5, 7, 11\}. The number 1 is not prime.
- \{1, 4, 9\}, since 1 = 1^2, 4 = 2^2 and 9 = 3^2, and 4^2 = 16 is too large.
- \varnothing, the impossible event. No card is greater than 12.
- \{\text{Tue}, \text{Thu}, \text{Sat}, \text{Sun}\}. Tuesday, Thursday, Saturday and Sunday contain a u; Monday, Wednesday and Friday do not.
- \{\text{red}, \text{green}, \text{yellow}\}. This is the event “not blue”.
- P and R. The 6 is even and a multiple of 3. It is not greater than 6, and it is not a square number.
- A roll of 2 makes both E and F happen. Only a roll of 3 makes none of them happen: the three events together hold 1, 2, 4, 5 and 6. No outcome is in both E and G, and an event happens only when the outcome is one of its members, so a single roll cannot make both happen.
- 8 events. Writing g, b and w for the colors: the impossible event \varnothing; the simple events \{\text{g}\}, \{\text{b}\}, \{\text{w}\}; then \{\text{g}, \text{b}\}, \{\text{g}, \text{w}\}, \{\text{b}, \text{w}\}; and the certain event \{\text{g}, \text{b}, \text{w}\} = S. Sorting by size makes sure none is missed.
- The number 5 is an outcome, the single result of the roll. The set \{5\} is the simple event “the cube shows a 5”, which happens exactly when the outcome is 5. Everyday speech uses “rolling a 5” for both; without braces it is the outcome, with braces the event.
- Every trial gives an outcome in S, and the certain event is S itself, so it always happens. The impossible event \varnothing has no members, so no outcome is one of them, and it never happens.
- Yes. The odd numbers on a cube greater than 1 are 3 and 5, so A = \{3, 5\}. The primes on a cube are 2, 3 and 5, and leaving out 2 gives B = \{3, 5\}. An event is the set of outcomes it holds, and these two sets have the same members, so A = B.