Multiplying probabilities
Use the multiplication rule and tree diagrams to find probabilities of combined outcomes for independent and dependent events.
Worked examples
Multiplying probabilities for independent events
Straightforward
Problem
A spinner has 3 equal sections labelled 1, 2 and 3. A fair coin is flipped at the same time as the spinner is spun. Find the probability that the spinner lands on 2 and the coin shows heads.
1
Check whether the two events are independent.
The result of the spinner cannot affect the result of the coin, and the coin cannot affect the spinner. The two events are **independent**.
2
Find the probability that the spinner lands on 2.
The spinner has 3 equal sections, so each section has an equal chance of being selected.
3
Find the probability that the coin shows heads.
A fair coin has two equally likely outcomes.
4
Apply the multiplication rule for independent events.
Answer
Tree diagram for a two-stage experiment
Moderate
Problem
A bag contains 3 red and 2 blue balls. One ball is drawn, its colour noted, and it is **replaced**. A second ball is then drawn. Draw a tree diagram and find the probability that both balls drawn are the same colour.
1
Set up the first level of the tree diagram.
There are balls in total. The two possible outcomes on the first draw are red and blue.
2
Set up the second level of the tree diagram.
Because the ball is replaced after the first draw, the bag always contains 3 red and 2 blue balls. The probabilities on the second draw are the same as on the first draw.
3
Find P(both red) by multiplying along the red–red branch.
4
Find P(both blue) by multiplying along the blue–blue branch.
5
Add the two probabilities to find P(same colour).
The outcomes RR and BB are mutually exclusive, so their probabilities are added.
Answer
At least one using the complement rule
Challenging
Problem
A bag contains 3 red and 5 blue marbles. Two marbles are drawn one at a time **without replacement**. Find the probability that at least one marble is red. Use the complement rule.
1
Identify the complement event.
"At least one red" means one red or two red marbles. Its complement is "no red marbles at all", which means both marbles drawn are blue. It is easier to calculate the complement probability.
2
Find the probability that the first marble is blue.
There are marbles in total, of which 5 are blue.
3
Find the probability that the second marble is blue, given the first was blue.
After one blue marble is removed, 3 red and 4 blue marbles remain — marbles in total.
4
Calculate P(both blue).
5
Apply the complement rule.
Answer
Practise
Q1·Straightforward
A spinner has 4 equal sections numbered 1, 2, 3 and 4. A fair coin is flipped at the same time as the spinner is spun. Find the probability that the spinner lands on 3 **and** the coin shows tails.
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Explanation
and . Since the spinner and coin are independent, .
Q2·Straightforward
A standard die is rolled and a spinner with 3 equal sections labelled red, blue and green is spun. Find the probability that the die shows a 6 **and** the spinner lands on red.
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Explanation
and . These events are independent, so .
Q3·Straightforward
Events and are independent. and . Find .
Explanation
Since and are independent, .
Q4·Straightforward
A bag contains 5 red and 5 blue marbles. One marble is drawn, its colour is noted, and it is **replaced**. A second marble is then drawn. Find the probability that both marbles are red.
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Explanation
for each draw. Since the marble is replaced, the draws are independent. .
Q5·Moderate
A bag contains 2 red and 3 blue balls. A ball is drawn, its colour noted, and then **replaced**. A second ball is then drawn. Using a tree diagram, find the probability that the first ball is red and the second ball is blue.
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Explanation
. Since the ball is replaced, . Multiplying along the red-blue branch of the tree diagram: .
Q6·Moderate
A fair die is rolled twice. Find the probability that the first roll shows a 5 **and** the second roll shows an odd number.
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Explanation
. The odd numbers on a die are 1, 3 and 5, so . .
Q7·Moderate
The probability that it rains on a given day is 0.3. If it rains, the probability that Maya takes an umbrella is 0.9. If it does not rain, the probability she takes an umbrella is 0.1. Using a tree diagram, find the probability that it rains **and** Maya takes an umbrella.
Explanation
The rain-and-umbrella branch has probability .
Q8·Moderate
The probability that Priya passes her driving test is 0.7. If she passes, the probability that she buys a car is 0.8. If she does not pass, the probability she buys a car is 0.2. Using a tree diagram, find the probability that Priya passes her test **and** buys a car.
Explanation
The pass-and-buy-car branch has probability .
Q9·Challenging
A bag contains 5 red and 3 blue marbles. Two marbles are drawn one at a time **without replacement**. Find the probability that both marbles are red. Give your answer as a fraction in simplest form.
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Explanation
. After removing one red marble, 4 red and 3 blue remain (7 total). . .
Q10·Challenging
A bag contains 4 yellow and 6 green balls. Two balls are drawn **without replacement**. Use the complement rule to find the probability that **at least one** yellow ball is drawn.
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Explanation
. After removing one green ball, 5 green remain from 9 total. . . .
Q11·Challenging
A standard deck has 52 cards, including 4 aces. Two cards are dealt one at a time **without replacement**. Find the probability that both cards are aces.
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Explanation
. After removing one ace, 3 aces remain from 51 cards. . .
Q12·Challenging
A group consists of 4 boys and 6 girls. Two students are chosen at random **without replacement**. Use the complement rule to find the probability that **at least one** girl is chosen.
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Explanation
. After removing one boy, 3 boys remain from 9 students. . . .