About this chapter
Probability questions in Statistics 1 are mostly Venn diagrams and tree diagrams with unknowns in them. The June 2022 examiners summed up the main weakness in one line: students often assume independence when a conditional probability should be used instead.
This chapter covers filling in Venn diagrams, the addition rule, mutually exclusive and independent events, conditional probability, tree diagrams, reversing a conditional, sequences of trials, and setting up equations for unknown regions. The later questions use independence to find unknown probabilities, reverse a medical-style test result, and link a probability back to a real count of people.
The ten questions
- Q1 (5 marks): a Venn diagram of students studying Art and Music: a probability, a conditional probability, and a test for independence.
- Q2 (5 marks): P(A and B) from the addition rule, then show A and B are independent.
- Q3 (5 marks): two counters taken without replacement on a tree diagram: same colour and at least one blue.
- Q4 (5 marks): independent events with P(A) = 3P(B) give a quadratic in p, with one root rejected.
- Q5 (5 marks): a Venn diagram of two independent events gives a quadratic in x, and a condition picks the root.
- Q6 (11 marks): three events F, G and K with independence and mutual exclusion, ending in a full Venn diagram of exact probabilities.
- Q7 (11 marks): a factory test that is not perfect: the chance an item is flagged, the chance a flagged item is really faulty, then a second test.
- Q8 (10 marks): a three-event Venn diagram with unknowns p and q, a test for independence, and an estimated count of people.
- Q9 (11 marks): a three-event Venn diagram with four unknowns found from P(A), P(A or B) and independence.
- Q10 (12 marks): rain and a late bus: show the chance of rain, reverse the conditional, then two days in a row.
Key skills tested
Venn diagrams. Fill them in from the middle outwards. Every region needs a value, including 0 and the region outside all the sets.
The addition rule. P(A or B) = P(A) + P(B) − P(A and B).
Mutually exclusive events. A and B cannot both happen, so P(A and B) = 0 and P(A or B) = P(A) + P(B).
Independence. P(A and B) = P(A) × P(B), or equivalently P(A | B) = P(A). Show the numbers when you test it.
Conditional probability. P(A | B) = P(A and B) ÷ P(B): restrict attention to B, then find the part that is also A.
Tree diagrams. Multiply along the branches and add across outcomes. Complete every branch before you start.
Reverse conditionals. P(first | second) = P(both) ÷ P(second), where P(second) adds every route to the second event.
Sequences of trials. Winning on the second turn needs every earlier failure too, so it is a product of several probabilities. Without replacement, the numbers change each time.
Equations for unknown regions. Write each given probability as a sum of regions, and use independence for a product. Solve simultaneously and check each answer is between 0 and 1.
Worked example
Question 2 from this chapter. P(A) = 0.4, P(B) = 0.35 and P(A or B) = 0.61. (a) Find P(A and B). (b) Show that A and B are independent. (c) Find P(A′ and B).
(a) By the addition rule, 0.61 = 0.4 + 0.35 − P(A and B), so P(A and B) = 0.14.
(b) P(A) × P(B) = 0.4 × 0.35 = 0.14, which equals P(A and B), so A and B are independent.
(c) P(A′ and B) is the part of B outside A: 0.35 − 0.14 = 0.21.
In part (b) the numbers must be shown and compared, and the conclusion written out.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2024, Q6: a Venn diagram where independence gives an unknown probability, a conditional probability, then a count of people from a probability, as in Q8 and Q9.
- June 2022, Q4: a Venn diagram with a given result to show, a conditional probability and a full diagram including an empty region, as in Q1 and Q6(c).
Where marks are lost
- Assuming independence. The June 2022 examiners noted that students often multiplied probabilities when a conditional probability was needed.
- A probability above 1. In June 2022 some gave conditional probabilities greater than one without any sign of doubt. A quick check catches this.
- Blank regions. A blank space in a Venn diagram is not read as zero. In June 2022 students lost marks for leaving out the 0 in the region outside the sets.
- The wrong region for a count. In June 2024 the most common error when converting to a number of people was linking the count to the wrong region of the diagram.
Common questions
How do I show two events are independent?
Work out P(A) × P(B) and P(A and B) as numbers, show that they are equal, and state the conclusion.
What is the difference between independent and mutually exclusive?
Mutually exclusive events cannot happen together. Independent events can, but one does not change the probability of the other.
How do I reverse a conditional probability?
Divide the probability of both events by the total probability of the event you are given, adding every route to it on the tree.
Where this chapter leads
- Statistics 1 Chapter 6: Discrete Random Variables: probability distributions and games built from several trials.
- Statistics 1 Chapter 7: The Normal Distribution: conditional probabilities within a normal model.