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A tree diagram shows all possible outcomes of two or more events, with each branch representing a possible result.
Each branch is labelled with its probability; probabilities on branches from the same node must sum to 1.
Multiply along a path: P(A then B) = P(A) × P(B | A). This becomes P(A) × P(B) only for independent events.
To find the probability of either of two distinct outcomes, add their probabilities (e.g. P(A then B) or P(B then A)).
The sum of probabilities of all final outcomes equals 1.
Two independent fair-coin tosses
Independent vs Dependent Events
Independent events: the outcome of one does not affect the other (e.g. tossing a coin twice).
For independent stages, the second-stage probability is unchanged by the first outcome. It does not have to equal a first-stage probability for a different event.
Dependent events: the outcome of the first event changes the probabilities for the second (e.g. drawing without replacement).
In without replacement problems, the second branch probabilities must be adjusted based on the first outcome.
Drawing Tree Diagrams
Start with a node and draw a branch for each possible outcome of the first event, writing the probability on each.
From each first branch, draw branches for the second event, writing the conditional probabilities (given the first outcome).
Label each final outcome clearly (e.g. R,R; R,B; B,R; B,B).
Check that the probabilities on branches from each node sum to 1.
Calculating Probabilities of Combined Events
For a specific combined outcome (e.g. first red, then blue), multiply the probabilities along that path.
For 'at least one' type questions, it is often easier to use the complement: P(at least one) = 1 − P(none).
When a condition can be satisfied by multiple paths (e.g. one red and one blue), add the probabilities of all those paths.
Always check that your final probability is between 0 and 1.
Without Replacement Problems
In without replacement, the total number of items decreases after the first draw, so the second probabilities change.
Example: from 5 red and 3 blue, P(R then B) = (5/8) × (3/7) — the denominator drops to 7.
The probability of the second event depends on the first outcome, so use conditional probabilities.
Tree diagrams are especially useful for tracking these changing probabilities.
Two counters without replacement
Conditional Probability (Higher Tier)
P(A given B) means the probability of A occurring given that B has already occurred.
From a tree diagram, P(A given B) = probability of the path ending in A and B divided by the total probability of B.
From a two-way table, P(A given B) = (number in both A and B) / (total number in B).
Do not confuse P(A given B) with P(B given A) — they are usually different.
Conditioning on B requires P(B) > 0. If A ∩ B spans several final paths, add their probabilities before dividing by P(B).
Using Two-Way Tables and Venn Diagrams
Two-way tables show frequencies of two events; use them to find conditional probabilities by focusing on the relevant row or column.
Venn diagrams can represent probabilities of events and their intersections; conditional probability is the intersection divided by the condition's probability.
These representations help visualise combined events and avoid errors.
Conditional probability from counts
Common Misconceptions and Errors
Forgetting to adjust second branch probabilities when there is no replacement.
Adding along a branch instead of multiplying — multiply for 'and', add for 'or'.
Missing routes for 'at least one' — include all possible paths that satisfy the condition.
Branch probabilities not summing to 1 at each node.
Confusing P(A given B) with P(B given A) — always check which event is the condition.
Typical Exam Question Forms
Complete a partially drawn tree diagram and find the probability of a combined event.
Find the probability of at least one occurrence (e.g. at least one head in two tosses).
Solve a without replacement problem (e.g. drawing two counters from a bag).
Find a conditional probability from a two-way table.
Form and solve an equation from a probability condition (e.g. find an unknown number of items).
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1.In a tree diagram, what is the sum of the probabilities on all branches coming from the same node?
Easy
A0
B0.5
C1
DIt depends on the situation
2.When multiplying probabilities along branches of a tree diagram, you are finding the probability of two events both happening.
Easy
True or false?
3.A fair coin is tossed twice. What is the probability of getting two heads?
Easy
A1/2
B1/4
C1/8
D1
4.A bag contains 4 green and 6 yellow counters. Two counters are taken at random without replacement. What is the probability that both are green?
Medium
A4/10 × 4/10
B4/10 × 3/9
C4/10 × 4/9
D4/10 × 3/10
5.Which of the following are correct statements about tree diagrams? (Select all that apply)
Medium
AThe probabilities on branches from a node must sum to 1.
BWhen events are dependent, the probabilities on the second set of branches may change.
CTo find the probability of one event OR another, you multiply along different branches.
DTo find the probability of a combined outcome, you multiply along the branches for that path.
6.When sampling without replacement, the probabilities on the second set of branches are the same as on the first set.
Easy
True or false?
7.Match each event type to the correct operation for finding the probability of the combined event.
Medium
Both A and B (independent)
Either A or B (mutually exclusive)
Both A and B (without replacement)
Multiply P(A) × P(B)
Add P(A) + P(B)
Multiply P(A) × P(B given A)
8.Put the steps in order to find the probability of at least one success in two independent trials.
Medium
Identify the no-success path through the two trials.
Subtract the no-success probability from 1.
Identify the success and failure probabilities for each independent trial.
Multiply the two failure probabilities to evaluate that path.
Mehr Fragen
9.Match each description to the correct probability rule.
Medium
P(A and B) for independent events
P(A or B) for mutually exclusive events
P(A and B) for dependent events
Multiply probabilities
Add probabilities
Multiply adjusted probabilities
10.Put the steps in order to find the probability of getting at least one head when tossing a coin twice.
Medium
List the outcomes with at least one head: HT, TH, HH
Multiply probabilities along each branch
Add the probabilities of the favourable outcomes
Draw a tree diagram with branches for each toss
11.A bag contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. Which of the following statements are correct? (Select all that apply)
Medium
AThe probability that the first marble is red is 4/10
BThe probability that the second marble is red given the first was red is 3/9
CThe probability that both marbles are red is 12/90
DThe probability that the second marble is blue given the first was red is 6/10
12.For independent stages of a probability tree, the second-stage probabilities do not depend on which first-stage outcome occurred.
Easy
True or false?
13.A bag contains 3 red and 5 green balls. Two balls are drawn without replacement. What is the probability that the balls are different colours?
Hard
A15/56
B15/28
C15/64
D30/64
14.A tree diagram shows two independent events. Which of the following are correct? (Select all that apply)
Hard
AThe sum of probabilities on branches from a node is 1
BThe probability of a combined outcome is found by adding along the branches
CThe probability of a combined outcome is found by multiplying along the branches
DThe probability of at least one event is 1 minus the probability that neither event occurs
15.Put the steps in order to find P(A given B) from a two-way table.
Medium
Restrict attention to the group satisfying B.
Within that group, count those who also satisfy A.
Divide that count by the total number satisfying B.
16.A bag contains 3 red and 5 blue beads. A bead is taken at random and not replaced, then a second bead is taken. What is the probability that both beads are red?
Medium
A3/8 × 2/7
B3/8 × 3/8
C3/8 × 2/8
D3/8 × 3/7
17.A fair coin is tossed twice independently. What is the probability of at least one head?
Medium
A1/4
B1/2
C3/4
D1
18.A bag contains 3 red and 5 green balls. Two balls are drawn without replacement. Which of the following probabilities are correct? (Select all that apply)
Medium
AP(two red) = 3/8 × 2/7
BP(two green) = 5/8 × 4/7
CP(one of each) = 3/8 × 5/7 + 5/8 × 3/7
DP(two red) = 3/8 × 3/8
19.Use the complement method for two independent tosses of a fair coin. Put these steps in order.
Medium
Identify the complementary event: no heads (two tails).
Subtract its probability from 1.
Multiply the two tail probabilities to evaluate the no-heads path.
Write down P(tail) = 1/2 for each independent toss.
20.A bag contains 3 red and 7 blue counters. Two counters are taken at random without replacement. What is the probability that both counters are blue?
Medium
A7/15
B49/100
C7/10
D21/50
21.A bag contains 5 red and 3 green balls. Two balls are drawn without replacement. What is the probability that both balls are the same colour?
Hard
A13/28
B5/8
C1/2
D25/64
22.A bag contains 4 red and 6 blue counters. Two counters are taken at random without replacement. Which of the following statements are correct? (Select all that apply)
Medium
AThe probability that both counters are red is 2/15.
BThe probability that both counters are blue is 1/3.
CThe probability that the counters are different colours is 8/15.
DThe probability that at least one counter is red is 2/3.
EThe probability that the first counter is red and the second is blue is 8/15.
23.When two events are independent, the probability that both occur is found by multiplying their individual probabilities.
Medium
True or false?
24.When drawing a tree diagram for sampling without replacement, the probabilities on the second set of branches are the same as those on the first set.
Medium
True or false?
25.A bag contains 5 red and 3 blue counters. Two counters are taken at random without replacement. What is the probability that the counters are of different colours?
Hard
A15/28
B15/32
C5/8
D3/8
26.Which of the following is the correct notation for the probability of event A given event B?
Easy
AP(A and B)
BP(A | B)
CP(B | A)
DP(A or B)
27.When sampling without replacement, the probabilities on the second set of branches of a tree diagram are different from those on the first set.
Medium
True or false?
28.A bag contains 3 red and 5 blue beads. A bead is taken at random and replaced, then a second bead is taken. What is the probability that both beads are red? Give your answer as a fraction.
Medium
A9/64
B3/8
C9/16
D3/64
29.A bag contains 5 green and 3 yellow counters. Two counters are taken at random without replacement. What is the probability that the two counters are of different colours?
Hard
A15/28
B15/56
C30/64
D5/8
30.Match each event type to the correct operation for finding the probability of the combined event on a tree diagram.
Medium
Event A and event B (both occur)
Event A or event B (either occurs)
Event A given event B (conditional)
Multiply probabilities along the branches
Add probabilities of the distinct branches
Use the branch probabilities from the relevant node
31.Put the steps in order to find the probability of getting at least one head when tossing a fair coin three times.
Medium
Draw a tree diagram showing all outcomes for three tosses.
Identify the branches that give at least one head.
Add the probabilities of all branches with at least one head.
Multiply probabilities along each branch to find the probability of that outcome.
32.Put the steps in order to find the probability of getting exactly one head when tossing a fair coin twice.
Medium
Identify the branches that give exactly one head (HT and TH).
Multiply probabilities along each of those branches.
Add the probabilities of the two branches.
Draw a tree diagram for the two tosses.
33.A bag contains 5 red and 3 blue counters. Two counters are taken at random without replacement. What is the probability that both counters are red?
Medium
A\frac{5}{8} \times \frac{4}{7}
B\frac{5}{8} \times \frac{5}{8}
C\frac{5}{8} \times \frac{3}{7}
D\frac{5}{8} \times \frac{4}{8}
34.When drawing a tree diagram for sampling without replacement, the probabilities on the second set of branches are different from those on the first set.
Easy
True or false?
35.Match each scenario to the correct type of events.
Medium
Tossing a coin twice
Drawing two cards without replacement
Rolling a die and tossing a coin
Independent events
Dependent events
36.A fair coin is tossed three times. Which calculation gives the probability of getting a head on the first toss, a tail on the second toss, and a head on the third toss?
37.In a tree diagram, the probabilities on branches that come from the same node must add up to 1.
Easy
True or false?
38.A bag contains 4 red and 6 blue counters. Two counters are taken at random without replacement. Which of the following are correct? (Select all that apply)
Medium
AP(first red) = \frac{4}{10}
BP(second red given first red) = \frac{4}{10}
CP(second blue given first red) = \frac{6}{9}
DP(both red) = \frac{4}{10} \times \frac{3}{9}
EP(both blue) = \frac{6}{10} \times \frac{6}{9}
39.Put the following steps in order to find the probability of getting at least one head when tossing a fair coin twice.
Medium
Find the probability of getting no heads: \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
Subtract from 1: 1 - \frac{1}{4} = \frac{3}{4}
Identify that 'at least one head' is the complement of 'no heads'
Write the answer: \frac{3}{4}
40.Match each scenario to the correct operation on a tree diagram.
Medium
Finding the probability of a single path (e.g., red then blue)
Finding the probability of either one path or another (e.g., red then blue OR blue then red)
Finding the probability of 'at least one' success
Multiply the probabilities along the path
Add the probabilities of the distinct paths
Subtract the probability of no success from 1
41.A bag contains 3 red and 5 blue beads. A bead is taken at random and replaced, then a second bead is taken. What is the probability that both beads are red? Give your answer as a fraction in its simplest form.
Medium
42.Complete the sentence.
Easy
When two events are independent, the probability of both happening is found by ____ the individual probabilities.
43.State the probability of an event that is certain to happen.
Medium
44.A box contains 5 red and 3 blue pens. Two pens are taken at random without replacement. What is the probability that the two pens are of different colours? Give your answer as a fraction in its simplest form.
Hard
45.A spinner has 3 equal sections labelled A, B and C. It is spun twice. What is the probability of getting A on the first spin and B on the second spin? Give your answer as a fraction in its simplest form.
Medium
46.A bag contains 5 red and 7 blue counters. Two counters are taken at random without replacement. Find the probability that the two counters are the same colour. Give your answer as a fraction in its simplest form.
Medium
47.A spinner has 4 equal sections labelled 1, 2, 3, 4. It is spun twice. Write down the probability of getting a 1 on the first spin and a 3 on the second spin as a fraction.
Medium
48.A bag contains 6 red and 4 blue counters. Two counters are taken at random without replacement. What is the probability that both counters are red? Give your answer as a decimal to 3 decimal places.
Medium
49.A bag contains 5 green and 3 yellow counters. Two counters are taken at random without replacement. What is the probability that the two counters are of different colours? Give your answer as a decimal to 3 decimal places.
Medium
50.Complete the sentence about the probabilities on a tree diagram.
Easy
The probabilities on all branches coming from the same node must ____ to 1.
51.Complete the rule for combining probabilities on a tree diagram.
Medium
When finding the probability of two events both happening, you ____ the probabilities along the branches.
52.Complete the rule for combining probabilities of alternative outcomes.
Medium
For two mutually exclusive final paths on a probability tree, you ____ their probabilities to find the probability of either path.
53.Explain how dependence changes second-stage branch probabilities compared with the unconditional probability of the second event.
Hard
54.Explain why, when using a tree diagram to find the probability of 'at least one' success, it is often easier to find the probability of no successes and subtract from 1.
Hard
55.Explain why, when sampling without replacement, the probabilities on the second set of branches of a tree diagram are different from those on the first set.
Hard
56.A bag contains 3 red and 5 blue beads. A bead is taken at random and replaced, then a second bead is taken. What is the probability that both beads are red? Give your answer as a fraction in its simplest form.
Medium
A\frac{9}{64}
B\frac{3}{8}
C\frac{9}{16}
D\frac{15}{64}
57.A box contains 5 red and 3 blue pens. Two pens are taken at random without replacement. What is the probability that the two pens are of different colours? Give your answer as a fraction in its simplest form.
Hard
A\frac{15}{56}
B\frac{5}{14}
C\frac{15}{28}
D\frac{3}{7}
58.A spinner has 3 equal sections labelled A, B and C. It is spun twice. What is the probability of getting A on the first spin and B on the second spin? Give your answer as a fraction in its simplest form.
Medium
A3 /9
B0 /9
C1 /9
D2 /9
59.A bag contains 6 red and 4 blue counters. Two counters are taken at random without replacement. What is the probability that both counters are red? Give your answer as a decimal to 3 decimal places.
Medium
A-0.667
B2.333
C1.333
D0.333
60.A bag contains 5 green and 3 yellow counters. Two counters are taken at random without replacement. What is the probability that the two counters are of different colours? Give your answer as a decimal to 3 decimal places.
Medium
A1.536
B0.536
C2.536
D-0.464
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