Year 10 · Mathematics · Probability
Two-step chance experiments
List the outcomes of two-step experiments using arrays and tree diagrams, including with and without replacement, and calculate probabilities.
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These lessons were written around Australian Curriculum strands and matched to UK year groups by age (Year 1 is the year a child turns 6 in England). They are not mapped to the National Curriculum in England or the curricula of Scotland, Wales or Northern Ireland, so check them against your own plans.
Learning goals
Learners will
- List the outcomes of two-step chance experiments using arrays (tables) and tree diagrams.
- Calculate probabilities from the sample space of a two-step experiment.
- Distinguish between selections made with and without replacement.
Success looks like
- I can draw a 6 × 6 array for rolling two dice and use it to find probabilities.
- I can draw a tree diagram and multiply along the branches.
- I can explain how the probabilities change on the second draw when an item is not replaced.
The big idea
In a two-step experiment there are two stages, such as rolling two dice or drawing two counters from a bag. To find probabilities, we need a complete list of outcomes — the sample space.
An array (table) works well when both steps have equally likely outcomes. For two dice, write 1–6 across the top and 1–6 down the side. Every cell is one outcome, so there are 6 × 6 = 36 equally likely outcomes. For a coin and a die there are 2 × 6 = 12 outcomes.
A tree diagram shows each step as a set of branches. Write the probability on each branch. To find the probability of one path, multiply the probabilities along the branches. To find the probability of an event made from several paths, add the probabilities of those paths. The probabilities of all the paths add to 1.
Order matters in listing outcomes: for two coins, HT (head then tail) and TH (tail then head) are different outcomes. So the sample space is {HH, HT, TH, TT}, and P(one head and one tail) = 24 = 12.
With replacement means the first item is put back, so the second draw is exactly like the first. Without replacement means the first item stays out, so on the second draw there is one fewer item in total — and one fewer of whatever was taken. The probabilities on the second set of branches change.
For example, with 3 red and 2 blue counters: with replacement, P(red then red) = 35 × 35 = 925. Without replacement, after a red is taken there are 2 red out of 4, so P(red then red) = 35 × 24 = 620 = 310.
Worked examples
Two dice in an array
Two dice are rolled and the numbers added. Find P(total of 7).
- Draw a 6 × 6 array: there are 36 equally likely outcomes.
- The pairs that total 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1) — 6 outcomes.
- P(total of 7) = 636 = 16.
Answer: 16
Tree diagram without replacement
A bag has 3 red and 2 blue counters. Two are taken without replacement. Find P(both red) and P(one of each colour).
- First draw: P(R) = 35, P(B) = 25.
- If red was taken first, 4 counters remain (2 red, 2 blue): P(R) = 24, P(B) = 24.
- If blue was taken first, 4 remain (3 red, 1 blue): P(R) = 34, P(B) = 14.
- P(RR) = 35 × 24 = 620 = 310.
- One of each: P(RB) + P(BR) = 35 × 24 + 25 × 34 = 620 + 620 = 1220 = 35.
Answer: P(both red) = 310; P(one of each) = 35
With replacement
Repeat the previous experiment, but put the first counter back before drawing again. Find P(both red).
- The bag is the same for both draws: P(R) = 35 each time.
- P(RR) = 35 × 35 = 925.
Answer: 925 (compared with 310 without replacement)
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
How many outcomes are in the sample space when two six-sided dice are rolled?
36 — Each die has 6 outcomes, so there are 6 × 6 = 36 combinations.
A coin is flipped and a six-sided die is rolled. How many outcomes are there?
12 — 2 × 6 = 12 outcomes, such as H1, H2, …, T6.
Two coins are flipped. What is P(two heads)?
14 — The sample space is {HH, HT, TH, TT}. Only HH is two heads: 14.
What does “without replacement” mean?
The item is not put back, so the second draw has one fewer item — Without replacement, the first item stays out, which changes the probabilities for the second draw.
Two dice are rolled and the numbers added. What is P(total of 10)? Give a fraction in simplest form.
1/12 — Totals of 10: (4, 6), (5, 5), (6, 4) — 3 of 36 outcomes. 336 = 112.
Two dice are rolled. What is the probability of rolling a double (both dice the same)? Give a fraction in simplest form.
1/6 — There are 6 doubles (1-1 to 6-6) out of 36 outcomes: 636 = 16.
A bag has 4 red and 6 green marbles. Two are drawn WITH replacement. What is P(both green)? Give a fraction in simplest form.
9/25 — P(G) = 610 each time. 610 × 610 = 36100 = 925.
The same bag (4 red, 6 green) is used, but two marbles are drawn WITHOUT replacement. What is P(both red)? Give a fraction in simplest form.
2/15 — 410 for the first red, then 3 red of the 9 left: 39. 410 × 39 = 1290 = 215.
Watch out for
- Counting HT and TH as the same outcome, giving P(one head, one tail) = 13. List the outcomes in order, or use two different coins, to see there are 4 outcomes.
- Adding along the branches of a tree instead of multiplying. Remember: multiply along a path, add between paths.
- Forgetting to change both the numerator and the denominator on the second draw without replacement.
Tips for parents
- Roll two dice 100 times and tally the totals. The learner will see 7 come up most often — then the array explains why.
- Use two different coloured dice so that outcomes such as (2, 5) and (5, 2) are clearly different.
- Act out “without replacement” with real counters in a bag, keeping the first one out on the table.
Go further
- Design a fair game using two dice, where each player has an equal chance of winning, and test it.
- Extend tree diagrams to three steps, such as flipping three coins (8 outcomes), and find P(exactly two heads) = 38.
- Investigate the chance that two people in a group share a birthday month, using with-replacement thinking.
About 50 minutes
You'll need
- Two dice and a coin
- A bag with coloured counters or marbles
- Paper, pencil and ruler
Key words
- Two-step experiment
- A chance experiment with two stages, such as flipping a coin twice or drawing two marbles.
- Array (table)
- A grid listing every combination of outcomes from the two steps.
- Tree diagram
- A branching diagram showing the outcomes of each step, with probabilities written on the branches.
- With replacement
- The first item is put back before the second is chosen, so the probabilities stay the same.
- Without replacement
- The first item is not put back, so there is one fewer item and the probabilities change.
Still tricky?
For parents
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