Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- A bag holds 3 red counters and 7 blue ones. What is P(red) as a decimal?
Answer
0.3. 3 out of 10 counters. - What is the probability of rolling a 7 on a standard die?
Answer
0. No face shows 7, so it is impossible. - A hall has 4 rows of 6 seats. How many seats are there?
Answer
24. 4 × 6, six in each of four rows.
Words used in this lesson
- sample space
- the list of every possible outcome. For example: H1, H2, H3, H4, T1, T2, T3, T4.
- outcome
- one possible result, taking every stage together. For example: tails and a 3.
- organised list
- a list with a system, so none is missed. For example: H1, H2, H3, then T1, T2, T3.
- two-way table
- a grid with one stage down the side and one along the top. For example: the dice grid.
- combination
- one particular pairing of choices. For example: a soup with a bread.
Every first choice meets every second choice
Kofi has 3 T-shirts and 4 caps. He wants to wear a different outfit every day for as long as he can. Before you open the figure, how many different outfits does he have: 7, 12, or something else?
The list of everything that could happen is called the sample space, and each single result in it is an outcome. To list a two-stage sample space without missing any, hold the first choice still and run through every second choice, then move to the next first choice.
An organised list or a table finds every outcome once. Keep the first result fixed (1A, 1B, 1C, …), then move on (2A, 2B, 2C, …). A table puts the first stage down the side and the second along the top.
Counting. When every first result can be paired with every second one, the number of outcomes is the number of first results times the number of second results. It still works when the first choice uses something up, as long as every first result leaves the same number of second results: 7 runners, then 6 left for second place, makes 7 × 6.
Slide each spinner, and say how many cells there will be before you look. Then set 3 numbers and 4 letters for Kofi's outfits.
2 numbers and 5 letters make 2 rows of 5 cells, so 2 × 5 = 10 outcomes: 1A, 1B, 1C, 1D, 1E, then 2A and so on.
Show me the steps
- 2 numbers and 5 letters: 2 rows of 5 cells, so 2 × 5 = 10 outcomes.
- 3 numbers and 3 letters: 3 rows of 3, so 9 outcomes.
- 4 numbers and 6 letters: 4 rows of 6, so 24 outcomes.
- 6 numbers and 6 letters: 36 outcomes, the same grid as two dice.
Multiply, do not add. 3 shirts and 4 caps is 3 × 4 = 12 outfits, not 3 + 4 = 7: each shirt goes with all 4 caps, so every shirt starts its own row of 4. Adding 3 + 4 only counts the separate items.
Worked examples
A coin is tossed and a 4-sided die is rolled. List every outcome.
- Hold H fixed and run the die: H1, H2, H3, H4.
- Then hold T fixed: T1, T2, T3, T4.
- That is 2 × 4 = 8 outcomes, and no pairing is missing or repeated.
Spinner 1 shows 1 to 4 and spinner 2 shows 1 to 3. In how many outcomes do the two numbers add to 5?
- Make the table of 4 rows and 3 columns: 12 cells.
- Pairs with a total of 5: (2, 3), (3, 2) and (4, 1).
- So 3 outcomes out of 12. (1, 4) is not there because spinner 2 has no 4.
A club chooses a president and a different secretary from 4 people, A, B, C and D. How many choices are there?
- Draw the 4 × 4 grid: president down the side, secretary along the top. That is 16 cells.
- Cross out the diagonal, AA, BB, CC and DD, because nobody can hold both jobs. The grid loses 4 cells.
- 16 − 4 = 12 choices. This is also 4 × 3: 4 people for president, and then 3 left for secretary. AB and BA both stay, because they are different jobs.
Your go. The first step is given. A 4-sided die (1 to 4) is rolled and a spinner with red, green and blue is spun. How many outcomes show an even number on the die?
- There are 4 × 3 = 12 outcomes in all.
- Finish it, then check.
Check the answer
The even numbers are 2 and 4, and each goes with 3 colours: 2 × 3 = 6 outcomes.
Check before you go on
1. A coin is tossed and a 6-sided die is rolled. How many outcomes are in the sample space?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A cafe offers 5 soups and 3 breads. How many different soup-and-bread choices are there?
3. One spinner shows the numbers 1 to 5 and another shows the numbers 1 to 4. How many outcomes are in the sample space when both are spun?
Level 2 · Building
4. Two 6-sided dice are rolled and the numbers added. In how many of the 36 outcomes is the total exactly 8?
5. Two 4-sided dice (numbered 1 to 4) are rolled. In how many outcomes are the two numbers different?
6. A coin is tossed and a spinner with red (R) and blue (B) is spun. Which list is complete, with no repeats?
Level 3 · Stretch
7. A cafe has some drinks and 4 snacks. There are 20 different drink-and-snack combinations. How many drinks are there?
8. A two-digit number is made from the digits 1 to 5, and the same digit cannot be used twice (so 33 is not allowed). How many such numbers are there?
Something to think about
Two spinners together have 12 outcomes in the grid. What could the two spinners look like? List every possibility, and say which ones are really the same grid turned on its side.
One way to think about it
2 × 6 and 3 × 4, and each one with the spinners swapped: 6 × 2 and 4 × 3 (a spinner with only 1 result is not really a spinner, so 1 × 12 is left out). Swapping the spinners turns the grid on its side, so 3 × 4 and 4 × 3 have the same 12 cells.
Exit ticket
9. A club chooses a president and then a secretary, who must be a different person, from Hana, Ivo and Jin. Which list shows every possible choice (president first)?