Outcomes and Sample Space

What a probability experiment is, and how to list all its possible outcomes as a sample space: with subscripts for identical objects, an ellipsis for a long list, and the total number of possible outcomes.

What You Should Already Know

From Primary school: some things are likely to happen and some are unlikely. This chapter turns such words into numbers, and it starts by listing everything that can happen.
Odd and even numbers, multiples, factors, prime numbers and perfect squares. For example, the multiples of 4 up to 20 are 4, 8, 12, 16 and 20, and the perfect squares up to 20 are 1, 4, 9 and 16.

Probability Experiments

Toss a coin and it lands on head or tail, but nobody can say which before it lands. Roll a die, spin a spinner or draw a name for a lucky draw, and the result again depends on chance. Activities like these are the starting point of probability.

Probability experiment and outcome

A probability experiment is an activity whose result cannot be predicted with certainty: the result depends on chance.

Each possible result of a probability experiment is called an outcome.

At random

An object is chosen at random when every object has the same chance of being chosen: the cards are shuffled or the balls mixed, and one is taken without looking.

Example 1: Is it a probability experiment?

Which of these are probability experiments? (a) Working out $5 + 7$ . (b) Tossing a coin to decide which team starts a game. (c) Measuring the length of your desk with a ruler. (d) Drawing one name at random from a box of 200 names in a lucky draw at the school carnival.

Solution

(a) $5 + 7$ is always 12. Nothing depends on chance.

(b) Head or tail cannot be predicted before the coin lands. It depends on chance.

(d) Any of the 200 names could be drawn, and nobody can say which. It depends on chance.

(b) and (d) are probability experiments; (a) and (c) are not.

The Sample Space

Sample space

The sample space of a probability experiment is the collection of all its possible outcomes.

We list the sample space between braces, $\{ \; \}$ , with the outcomes separated by commas. Each outcome is listed exactly once.

The number of outcomes in the list is the total number of possible outcomes.

Example 2: A coin, a die and a spinner

Write down the sample space and the total number of possible outcomes when (a) a coin is tossed; (b) a die is rolled and the number on its top face is noted; (c) the spinner below, with five equal sectors, is spun once.

Solution

(a) A coin lands showing a head (H) or a tail (T).

Sample space $= \{H, T\}$ . Total number of possible outcomes $= 2$ .

(b) A die has six faces, with 1 to 6 dots.

Sample space $= \{1, 2, 3, 4, 5, 6\}$ . Total number of possible outcomes $= 6$ .

Sample space = {Red, Orange, Blue, Green, Purple}. Total number of possible outcomes $= 5$ .

(a) $\{H, T\}$ , 2 outcomes; (b) $\{1, 2, 3, 4, 5, 6\}$ , 6 outcomes; (c) {Red, Orange, Blue, Green, Purple}, 5 outcomes.

Example 3: Numbered cards

Ten cards numbered 11, 12, 13, …, 20 are placed in a box. One card is drawn at random. Write down the sample space and the total number of possible outcomes.

Solution

Every number from 11 to 20 is on one card:

Sample space $= \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}\text{.}$

Counting the list: 10 outcomes.

Sample space $= \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}\text{;}$ total number of possible outcomes $= 10$ .

Identical Objects: Subscripts

Two black balls of the same size are identical: we cannot tell them apart by looking. They are still two different balls, and either one can be drawn. To count every outcome, we give each ball its own name with a small number, a subscript: $B_1$ and $B_2$ (read "B one" and "B two").

Example 4: Balls in a bag

A bag contains 2 identical black balls and 3 identical white balls. One ball is drawn at random. Write down the sample space and the total number of possible outcomes.

Solution

Let $B_1$ and $B_2$ stand for the two black balls, and $W_1$ , $W_2$ and $W_3$ for the three white balls.

Sample space $= \{B_1, B_2, W_1, W_2, W_3\}\text{.}$

Sample space $= \{B_1, B_2, W_1, W_2, W_3\}\text{;}$ total number of possible outcomes $= 5$ .

Why not {Black, White}?

Writing the sample space of Example 4 as {Black, White} gives 2 outcomes and hides the fact that there are more white balls than black ones.

Each of the 5 balls has the same chance of being drawn, so there are 5 outcomes, each as likely as the others: 2 of them are black and 3 are white. Drawing a white ball is more likely than drawing a black one. Lesson 13.2 turns this into numbers, and it needs all 5 outcomes.

Example 5: The letters of a word

Each letter of the word BANANA is written on a card, and the six cards are placed in a bag. One card is drawn at random. Write down the sample space and the total number of possible outcomes.

Solution

The word has 6 letters: one B, three As and two Ns. The three A cards are different cards, and so are the two N cards.

Name them in the order they appear in the word: B, $A_1$ , $N_1$ , $A_2$ , $N_2$ , $A_3$ . The B needs no subscript, as there is only one.

Sample space $= \{B, A_1, N_1, A_2, N_2, A_3\}\text{.}$

Sample space $= \{B, A_1, N_1, A_2, N_2, A_3\}\text{;}$ total number of possible outcomes $= 6$ .

Long Lists: the Ellipsis

Writing a long sample space

When the outcomes follow a clear pattern, write the first few to show the pattern, then three dots "…" (an ellipsis), then the last outcome: $\{10, 11, 12, \ldots, 99\}$ .

Always end with the last outcome, so that the reader knows where the list stops.

There are $b - a + 1$ whole numbers from $a$ to $b$ , both included.

Example 6: Two-digit numbers

A two-digit whole number is chosen at random. Write down the sample space and find the total number of possible outcomes.

Solution

The smallest two-digit number is 10 and the largest is 99.

Sample space $= \{10, 11, 12, \ldots, 99\}\text{.}$

Method 1: the numbers from 1 to 99 number 99; take away the 9 one-digit numbers 1 to 9: $99 - 9 = 90$ .

Method 2: from 10 to 99, both included: $99 - 10 + 1 = 90$ .

Note that $99 - 10 = 89$ is one short: it leaves out one end of the list.

Sample space $= \{10, 11, 12, \ldots, 99\}\text{;}$ total number of possible outcomes $= 90$ .

Picking Out Outcomes from a Sample Space

Often we are interested in only some of the outcomes: the even numbers, say, or the white balls. We find them by going through the sample space and picking out those that fit. Lesson 13.2 calls such a group of outcomes an event.

Example 7: Outcomes with a property

Twenty discs numbered 1 to 20 are placed in a box and one is drawn at random. (a) Write down the sample space. List the outcomes that are (b) multiples of 3; (c) prime numbers; (d) perfect squares.

Solution

(a) Sample space $= \{1, 2, 3, \ldots, 20\}$ , with $20 - 1 + 1 = 20$ outcomes.

(b) Multiples of 3 up to 20: 3, 6, 9, 12, 15, 18. That is 6 outcomes.

(c) A prime number has exactly two factors, 1 and itself (so 1 is not prime): 2, 3, 5, 7, 11, 13, 17, 19. That is 8 outcomes.

(d) Perfect squares up to 20: $1 = 1^2$ , $4 = 2^2$ , $9 = 3^2$ , $16 = 4^2$ . That is 4 outcomes ( $5^2 = 25$ is too big).

(a) $\{1, 2, 3, \ldots, 20\}$ ; (b) 3, 6, 9, 12, 15, 18; (c) 2, 3, 5, 7, 11, 13, 17, 19; (d) 1, 4, 9, 16.

Spot the Error

Example 8: Two wrong sample spaces

(a) A spinner has 8 equal sectors numbered 1 to 8. Mei Ling writes the sample space as $\{1, 2, 3, 4, 5, 6, 8\}$ . (b) A box holds 4 identical red counters and 1 green counter, and one counter is taken at random. Arjun writes the sample space as {Red, Green}. Find each mistake and write the correct sample space.

Solution

(a) The sector numbered 7 is missing: her list has 7 outcomes, but the spinner has 8 sectors. Check a list by counting it against the object.

Correct: $\{1, 2, 3, 4, 5, 6, 7, 8\}\text{,}$ 8 outcomes.

(b) The 4 red counters are 4 different counters. Listing "Red" once counts them as one outcome and hides that red is far more likely than green.

Correct: with $R_1$ to $R_4$ for the red counters and $G$ for the green one, $\{R_1, R_2, R_3, R_4, G\}$ , 5 outcomes.

(a) 7 is missing: $\{1, 2, 3, 4, 5, 6, 7, 8\}\text{.}$ (b) Identical counters must be told apart: $\{R_1, R_2, R_3, R_4, G\}$ .

Common Mistakes

Listing colours instead of objects. A bag of 2 black and 3 white balls has 5 outcomes, $\{B_1, B_2, W_1, W_2, W_3\}$ , not 2.
Leaving out an outcome or writing one twice. Count the list against the object: a die has 6 faces, so its sample space has 6 outcomes.
Counting the numbers from 10 to 99 as $99 - 10 = 89$ . Both ends are included: $99 - 10 + 1 = 90$ .
Ending a list with "…" and no last outcome, as in $\{1, 2, 3, \ldots\}$ , which reads as a list that never stops.

Practice

Try each question first, then open the solution to check your working.

Question 1

Which of these are probability experiments? Give a reason for each. (a) Rolling a die in a board game. (b) Counting the number of days in May. (c) Picking one ticket without looking from a box of lucky-draw tickets. (d) Weighing a 5 kg bag of rice on a kitchen scale.

ShowHide solution

(a) The number on the die cannot be predicted: chance.

(b) May always has 31 days: no chance.

(d) The bag has one mass; careful weighing gives the same reading each time: no chance.

(a) and (c) are probability experiments; (b) and (d) are not.

Question 2

Five cards show the vowels A, E, I, O and U, one on each card. One card is drawn at random. Write down the sample space and the total number of possible outcomes.

ShowHide solution

Each vowel is on one card, so each card is one outcome.

Sample space $= \{A, E, I, O, U\}$ .

Sample space $= \{A, E, I, O, U\}$ ; total number of possible outcomes $= 5$ .

Question 3

A spinner has 6 equal sectors coloured red, blue, red, green, red and blue, as shown. It is spun once. Write down the sample space, telling the sectors of the same colour apart, and the total number of possible outcomes. How many of the outcomes are red?

ShowHide solution

The three red sectors are different sectors, as are the two blue ones. Name them $R_1$ , $R_2$ , $R_3$ and $B_1$ , $B_2$ ; the green sector is $G$ .

Sample space $= \{R_1, B_1, R_2, G, R_3, B_2\}\text{:}$ 6 outcomes.

Red outcomes: $R_1$ , $R_2$ , $R_3$ .

Sample space $= \{R_1, B_1, R_2, G, R_3, B_2\}\text{;}$ 6 possible outcomes, 3 of them red.

Question 4

Five identical blue marbles and four identical red marbles are placed in a bag. One marble is drawn at random. (a) Write down the sample space, using subscripts. (b) State the total number of possible outcomes.

ShowHide solution

(a) Let $B_1$ to $B_5$ stand for the blue marbles and $R_1$ to $R_4$ for the red marbles.

Sample space $= \{B_1, B_2, B_3, B_4, B_5, R_1, R_2, R_3, R_4\}\text{.}$

(b) $5 + 4 = 9$ outcomes.

(a) $\{B_1, B_2, B_3, B_4, B_5, R_1, R_2, R_3, R_4\}\text{;}$ (b) 9.

Question 5

Each letter of the word STATISTICS is written on a card, and one card is drawn at random. (a) Write down the sample space. (b) How many possible outcomes are there? (c) How many of the outcomes are the letter T? (d) How many are vowels?

ShowHide solution

The 10 letters, in order: S, T, A, T, I, S, T, I, C, S. There are three Ss, three Ts, two Is, one A and one C.

(a) Sample space $= \{S_1, T_1, A, T_2, I_1, S_2, T_3, I_2, C, S_3\}\text{.}$

(b) 10 outcomes, one for each card.

(d) The vowels are A, $I_1$ and $I_2$ : 3 outcomes.

(a) $\{S_1, T_1, A, T_2, I_1, S_2, T_3, I_2, C, S_3\}\text{;}$ (b) 10; (c) 3; (d) 3.

Question 6

At a school carnival, lucky-draw tickets numbered 100 to 250 are sold, one number on each ticket, and every ticket is sold. One ticket is drawn at random for the first prize. Write down the sample space using an ellipsis, and find the total number of possible outcomes.

ShowHide solution

Sample space $= \{100, 101, 102, \ldots, 250\}\text{.}$

From 100 to 250, both included: $250 - 100 + 1 = 151$ .

Sample space $= \{100, 101, 102, \ldots, 250\}\text{;}$ total number of possible outcomes $= 151$ .

Question 7

Fifteen discs numbered 1 to 15 are placed in a bag and one is drawn at random. List the outcomes that are (a) even numbers; (b) multiples of 4; (c) factors of 12; (d) prime numbers. Say how many there are in each case.

ShowHide solution

Sample space $= \{1, 2, 3, \ldots, 15\}$ .

(a) Even: 2, 4, 6, 8, 10, 12, 14. That is 7.

(b) Multiples of 4: 4, 8, 12. That is 3 (16 is past 15).

(d) Prime: 2, 3, 5, 7, 11, 13. That is 6.

(a) 2, 4, 6, 8, 10, 12, 14 (7); (b) 4, 8, 12 (3); (c) 1, 2, 3, 4, 6, 12 (6); (d) 2, 3, 5, 7, 11, 13 (6).

Question 8

A three-digit whole number is chosen at random. Write down the sample space and find the total number of possible outcomes.

ShowHide solution

The smallest three-digit number is 100 and the largest is 999.

Sample space $= \{100, 101, 102, \ldots, 999\}\text{.}$

From 100 to 999, both included: $999 - 100 + 1 = 900$ . (Or: 999 numbers from 1 to 999, less the 99 numbers from 1 to 99: $999 - 99 = 900$ .)

Sample space $= \{100, 101, 102, \ldots, 999\}\text{;}$ total number of possible outcomes $= 900$ .

Question 9

A class chooses its monitor by drawing one name at random from a box. The box holds the names of 4 girls, Aisha, Bee Lin, Chloe and Divya, and 3 boys, Farid, Ganesh and Jun Jie. Ravi writes the sample space as {girl, boy}. Explain his mistake and write the correct sample space.

ShowHide solution

Each of the 7 names can be drawn, so each name is an outcome. "Girl" stands for 4 different names and "boy" for 3.

His list has 2 outcomes and hides that a girl is more likely to be chosen than a boy.

Sample space = {Aisha, Bee Lin, Chloe, Divya, Farid, Ganesh, Jun Jie}: 7 outcomes.

He listed groups, not names. The sample space is {Aisha, Bee Lin, Chloe, Divya, Farid, Ganesh, Jun Jie}, with 7 possible outcomes.

Question 10

Cards numbered 1 to 20 are placed in a box and one is drawn at random. Wei Jie writes the sample space as $\{1, 2, 3, \ldots\}$ and says there are $20 - 1 = 19$ possible outcomes. Find his two mistakes.

ShowHide solution

His list does not end with the last outcome, so it reads as a list that never stops. It should be $\{1, 2, 3, \ldots, 20\}$ .

Both 1 and 20 are outcomes, so there are $20 - 1 + 1 = 20$ outcomes, not 19.

The list must end with 20: $\{1, 2, 3, \ldots, 20\}$ ; and there are 20 possible outcomes, not 19.

Key Points

A probability experiment is an activity whose result depends on chance; each possible result is an outcome.
The sample space is the collection of all possible outcomes, listed once each between braces: $\{H, T\}$ .
The total number of possible outcomes is the number of outcomes in the sample space.
Identical objects are still different outcomes: tell them apart with subscripts, $B_1, B_2, W_1, W_2, W_3$ .
A long list with a pattern is written with an ellipsis and its last outcome: $\{10, 11, 12, \ldots, 99\}$ .
There are $b - a + 1$ whole numbers from $a$ to $b$ .
To find the outcomes with a property, go through the sample space and pick out those that fit.