Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- A spinner has 5 sections and a coin is tossed. How many outcomes are there?
Answer
10. 5 × 2: every section pairs with heads and with tails. - Simplify 4/16.
Answer
1/4. Divide the top and the bottom by 4. - Work out 3 × 2 × 2.
Answer
12. 3 × 2 = 6, and 6 × 2 = 12.
Words used in this lesson
- tree diagram
- a diagram that branches at each stage. For example: the coin tree.
- branch
- one result at one stage. For example: H from a toss.
- path
- one branch from every stage, joined, which is one outcome. For example: H then T then H.
- stage
- one step of a multi-step event. For example: the second toss.
- at least
- this many or more. For example: at least one head.
Each path is an outcome
Two coins are tossed. Rafa says there are three outcomes: two heads, two tails, and one of each. Before you open the figure, is he right?
A tree diagram settles it. Each toss is a stage, drawn as a split into branches, and each branch is a result. Following one branch from every stage, left to right, is a path, and a path is exactly one outcome.
One complete path is one outcome. The number of outcomes is the number of branches at the first stage times the number at each later stage: three tosses of a coin make 2 × 2 × 2 = 8.
Label every stage. Call them the first coin and the second coin, even if they look the same. A path records what each stage gave, so a different order is a different path.
Set 2 tosses and look for 1 head. Then try 3 tosses and each number of heads from 0 to 3, and say how many paths you expect before you check.
3 tosses make 2 × 2 × 2 = 8 outcomes. Exactly 3 heads is on 1 path: HHH.
Show me the steps
- 1 toss, 1 head: 2 outcomes in all (H, T), and H is the 1 that matches.
- 2 tosses, 1 head: 4 outcomes (HH, HT, TH, TT). HT and TH both have exactly one head: 2 of the 4.
- 3 tosses, 0 heads: 8 outcomes, and only TTT has no heads: 1 of the 8.
- 3 tosses, 3 heads: 8 outcomes, and only HHH has three heads: 1 of the 8.
Do not merge paths that use the same results in a different order. Listing “two heads, two tails, one of each” counts kinds of result, not outcomes. Two heads is one path (HH) and two tails is one path (TT), but “one of each” is two paths, HT and TH. The four paths are equally likely; Rafa's three kinds are not. Toss two coins many times and “one of each” comes up about twice as often as “two heads”. Lesson 4 turns that into 1/2 against 1/4.
Worked examples
A cafe lets you choose 2 mains, 3 sides and 2 drinks, one of each. How many different meals?
- Three stages: 2 branches, then 3 from each, then 2 from each of those.
- Paths: 2 × 3 × 2 = 12 meals.
A bag holds a red (R) counter and a blue (B) counter, and a spinner shows 1, 2 or 3. A counter is picked and the spinner spun. How many outcomes show red and an odd number?
- The tree has 2 first branches (R, B), each with 3 second branches (1, 2, 3): 6 paths.
- Red paths with an odd number: R1 and R3, so 2 outcomes out of 6.
Your go. The first step is given. The first stage gives A or B, and the second gives 1, 2 or 3. How many outcomes show B, or a 3, or both?
- The 6 outcomes are A1, A2, A3, B1, B2, B3.
- Finish it, then check.
Check the answer
B1, B2, B3 and A3: 4 outcomes. B3 has both, and it counts once.
Check before you go on
1. A coin is tossed three times. How many outcomes are there?
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 tree diagram has 4 branches at the first stage, and each of those has 3 branches at the second stage. How many complete paths are there?
3. Two coins are tossed. How many outcomes have at least one head?
Level 2 · Building
4. A family has 3 children. Treat each child as equally likely to be a girl (G) or a boy (B), and count the order of birth. In how many of the outcomes are there at least two girls?
5. A spinner has three equal sections numbered 1, 2 and 3. It is spun twice and the two numbers added. In how many outcomes is the total 4?
6. A coin is tossed and then a 4-sided die (1 to 4) is rolled. Which statement about the tree diagram is correct?
Level 3 · Stretch
7. A pizza shop sells 3 sizes and 2 crusts, with some number of toppings. There are 24 different pizzas in all. How many toppings are there?
8. A code has three characters. The first must be the digit 1, the second is one of the letters A, B or C, and the third is one of the digits 5, 6, 7 or 8. How many codes are there?
Something to think about
A coin is tossed 4 times. Without drawing the tree, how many outcomes are there, and how many of them have exactly 2 heads? Explain how the 3-toss tree tells you the first number.
One way to think about it
16 outcomes, because every extra toss doubles the paths: 2 × 2 × 2 × 2. Exactly 2 heads is on 6 of them: HHTT, HTHT, HTTH, THHT, THTH, TTHH. The 3-toss tree has 8 paths, and each one splits in two when a 4th toss is added.
Exit ticket
9. A spinner has 5 equal sections of different colours. It is spun twice. In how many outcomes does it land on the same colour both times?