Multi-stage experiments
List sample spaces and count favourable outcomes for two-stage experiments; find probability using tables of outcomes; calculate probability for experiments without replacement.
Worked examples
Listing outcomes for a two-stage experiment
Straightforward
Problem
A coin is tossed and a spinner with 3 equal sectors labelled 1, 2 and 3 is spun. List all possible outcomes and state the total number.
1
List the outcomes for each stage.
Coin: H or T (2 outcomes). Spinner: 1, 2 or 3 (3 outcomes).
2
Form all ordered pairs systematically.
3
Count the total number of outcomes.
outcomes in total.
Answer
6 outcomes: .
Finding probability from a table of outcomes
Moderate
Problem
Two fair 6-sided dice are rolled. Find .
1
Count the total number of equally likely outcomes.
equally likely outcomes.
2
List all favourable outcomes where the sum equals 8.
— 5 pairs.
3
Write the probability as a fraction.
Answer
Probability without replacement
Challenging
Problem
A bag contains 3 red and 2 blue marbles. Two marbles are drawn without replacement. Find .
1
Find the probability that the first marble drawn is blue.
There are 2 blue marbles out of 5 total.
2
Find the probability that the second marble is blue, given the first was blue.
After removing one blue marble, 4 marbles remain with 1 blue.
3
Multiply the two probabilities.
Answer
Practise
Q1·Straightforward
A fair coin is tossed twice. How many equally likely outcomes are in the sample space?
Explanation
The outcomes are HH, HT, TH and TT. There are equally likely outcomes in the sample space.
Q2·Straightforward
A fair coin is tossed and a standard 6-sided die is rolled. How many outcomes are in the sample space?
Explanation
The coin has 2 outcomes (H or T) and the die has 6 outcomes (1–6). The total number of outcomes is .
Q3·Straightforward
A spinner with 3 equal sectors labelled 1, 2 and 3 is spun and a fair coin is tossed. How many outcomes are in the sample space?
Explanation
The spinner has 3 outcomes and the coin has 2 outcomes. The total number of outcomes is : .
Q4·Straightforward
A fair coin is tossed twice. The sample space is . How many outcomes contain exactly one Head?
Explanation
The outcomes with exactly one Head are HT and TH. There are 2 such outcomes.
Q5·Moderate
Two fair 6-sided dice are rolled. Find . Give your answer as a fraction in simplest form.
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Explanation
There are 36 equally likely outcomes. Pairs that sum to 7: — 6 pairs.
Q6·Moderate
Two fair 6-sided dice are rolled. Find . Give your answer as a fraction in simplest form.
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Explanation
Pairs that sum to 5: — 4 pairs.
Q7·Moderate
Two fair 6-sided dice are rolled. Find the probability that both dice show the same number. Give your answer as a fraction in simplest form.
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Explanation
Outcomes where both dice match: — 6 pairs.
Q8·Moderate
Two fair 6-sided dice are rolled. Find . Give your answer as a fraction in simplest form.
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Explanation
.
Q9·Challenging
A bag contains 3 red and 2 blue marbles. One marble is drawn and not replaced, then a second marble is drawn. Find . Give your answer as a fraction in simplest form.
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Explanation
. After removing one red marble, 4 marbles remain with 2 red. .
Q10·Challenging
A bag contains 4 white and 2 black counters. Two counters are drawn without replacement. Find . Give your answer as a fraction in simplest form.
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Explanation
. After removing one black counter, 5 counters remain with 1 black. .
Q11·Challenging
A bag contains 3 red and 3 blue marbles. Two marbles are drawn without replacement. Find . Give your answer as a fraction in simplest form.
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Explanation
. .
Q12·Challenging
A box contains 5 red and 5 green cards. One card is drawn and not replaced, then a second card is drawn. Compared to drawing with replacement, what happens to ?
Explanation
With replacement: . Without replacement: . Since , the probability decreases — there is one fewer red card available for the second draw.